This table is a draft analysis for my book; at the moment, it contains many inaccuracies and incorrect analysis errors.


repeat ~as many times [ e.g. n, ~3 = n,n,n ]
insert...insert ~how many attachments [ e.g. n(...n...)n ~3 = n(n(n(n)n)n)n ]


2000 steps

1FGH
operation
hyperoperator
array notation
0
n+1
n[1]1
{n}+1
2FGH
FGH Veblen φ
FGH ψ-function
operation
hyperoperator
array notation
1
φ(0) = φ(00)
ψ(0)
2×n
n[1]n = 2[2]n
{n}×2
3FGH
FGH Veblen φ
FGH ψ-function
operation
hyperoperator
array notation
2
φ(0)+φ(0)
ψ(0)+ψ(0)
2n×n
2[3]n[2]n
{2,n}×n
4FGH
FGH Veblen φ
FGH ψ-function
operation
Conway notation

hyperoperator
array notation
3
φ(0)+φ(0)+φ(0)
ψ(0)+ψ(0)+ψ(0)
2↑↑n
2→n→2
2[4]n
{2,n,2}
5FGH
FGH Veblen φ
FGH ψ-function
operation
Conway notation

hyperoperator
array notation
4
φ(0)+φ(0)+φ(0)+φ(0)
ψ(0)+ψ(0)+ψ(0)+ψ(0)
2↑↑↑n
2→n→3
2[5]n
{2,n,3}
6FGH
FGH Veblen φ
FGH ψ-function
operation
Conway notation

hyperoperator
array notation
5
φ(0)+φ(0)+φ(0)+φ(0)+φ(0)
ψ(0)+ψ(0)+ψ(0)+ψ(0)+ψ(0)
2↑↑↑↑n
2→n→4
2[6]n
{2,n,4}
7name
FGH

FGH Veblen φ
FGH ψ-function
operation
Conway notation

hyperoperator
array notation
First transfinite ordinal
ω
φ(1) = φ(0)+φ(0)+...  = φ(φ(0)) = φ(10)
ψ(1) = ψ(0)+ψ(0)+...  = ψ(ψ(0))
2↑n+3n
2→n→n+3 < n→n→n
2[n+1]n  < n[n]n
{2,n,n+3} < {n,n,n} = {n,n,1,2}
8FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω+1
φ(1)+φ(0)
ψ(1)+ψ(0)
n→n→n→2
n[n,n]n = n[...n...]n ~n insert
{n,n,2,2} = {n,n,...{n,n,n}...} ~n-1 insert
9FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω+2
φ(1)+φ(0)+φ(0)
ψ(1)+ψ(0)+ψ(0)
n→n→n→3
n[n,n,n]n
{n,n,3,2}
10FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω+3
φ(1)+φ(0)+φ(0)+φ(0)
ψ(1)+ψ(0)+ψ(0)+ψ(0)
n→n→n→4
n[n,n,n,n]n
{n,n,3,2}
11FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω+ω = ω×2
φ(1)+φ(0)+φ(0)+... = φ(1)+φ(1)
ψ(1)+ψ(0)+ψ(0)+... = ψ(1)+ψ(1)
n→n→n→n→2
n[[n,n]]n = n[n,]n ~n-1 repeat
{n,n,1,3} = {n,n,n,2}
12FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×2+1
φ(1)+φ(1)+φ(0)
ψ(1)+ψ(1)+ψ(0)
n→n→n→n→3
n[[n,n,n]]n
{n,n,2,3}
13FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×2+2
φ(1)+φ(1)+φ(0)+φ(0)
ψ(1)+ψ(1)+ψ(0)+ψ(0)
n→n→n→n→4
n[[n,n,n,n]]n
{n,n,3,3}
14FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×2+3
φ(1)+φ(1)+φ(0)+φ(0)+φ(0)
ψ(1)+ψ(1)+ψ(0)+ψ(0)+ψ(0)
n→n→n→n→5
n[[n,n,n,n,n]]n
{n,n,4,3}
15FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×3
φ(1)+φ(1)+φ(0)+φ(0)+... = φ(1)+φ(1)+φ(1)
ψ(1)+ψ(1)+ψ(0)+ψ(0)+... = ψ(1)+ψ(1)+ψ(1)
n→n→n→n→n→2
n[[[n,n]]]n = n[[n,]]n ~n-1 repeat
{n,n,1,4} = {n,n,n,3}
16FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×3+1
φ(1)+φ(1)+φ(1)+φ(0)
ψ(1)+ψ(1)+ψ(1)+ψ(0)
n→n→n→n→n→3
n[[[n,n,n]]]n
{n,n,2,4}
17FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×3+2
φ(1)+φ(1)+φ(1)+φ(0)+φ(0)
ψ(1)+ψ(1)+ψ(1)+ψ(0)+ψ(0)
n→n→n→n→n→4
n[[[n,n,n,n]]]n
{n,n,3,4}
18FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×3+3
φ(1)+φ(1)+φ(1)+φ(0)+φ(0)+φ(0)
ψ(1)+ψ(1)+ψ(1)+ψ(0)+ψ(0)+ψ(0)
n→n→n→n→n→5
n[[[n,n,n,n,n]]]n
{n,n,4,4}
19FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×4
φ(1)+φ(1)+φ(1)+φ(0)+φ(0)+... = φ(1)+φ(1)+φ(1)+φ(1)
ψ(1)+ψ(1)+ψ(1)+ψ(0)+ψ(0)+... = ψ(1)+ψ(1)+ψ(1)+ψ(1)
n→n→n→n→n→n→2
n[[[[n,n]]]]n = n[[[n,]]]n ~n-1 repeat
{n,n,1,5} = {n,n,n,4}
20FGH
FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
ω×5
φ(1)+φ(1)+φ(1)+φ(1)+φ(0)+φ(0)+... = φ(1)+φ(1)+φ(1)+φ(1)+φ(1)
ψ(1)+ψ(1)+ψ(1)+ψ(1)+ψ(0)+ψ(0)+... = ψ(1)+ψ(1)+ψ(1)+ψ(1)+ψ(1)
n→n→n→n→n→n→n→2
n[[[[[n,n]]]]]n = n[[[[n,]]]]n ~n-1 repeat
{n,n,1,6} = {n,n,n,5}
21proof theory
FGH

FGH Veblen φ
FGH ψ-function
Conway notation
hyperoperator
array notation
RFA
ω×ω = ω2
φ(2) = φ(1)+φ(1)+φ(1)+...
ψ(2) = ψ(1)+ψ(1)+ψ(1)+...
n→ ~n+2 repeat
n[...n,n...]n ~n insert
{n,n,1,1,2} = {n,n,n,n}
22FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2+1
φ(2)+φ(0)
ψ(2)+ψ(0)
{n,n,2,1,2}
23FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2
φ(2)+φ(1)
ψ(2)+ψ(1)
{n,n,1,2,2}
24FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2+ω+1
φ(2)+φ(1)+φ(0)
ψ(2)+ψ(1)+ψ(0)
{n,n,2,2,2}
25FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2×2
φ(2)+φ(2)
ψ(2)+ψ(2)
{n,n,1,1,3} = {n,n,n,n,2}
26FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2×2+1
φ(2)+φ(2)+φ(0)
ψ(2)+ψ(2)+ψ(0)
{n,n,2,1,3}
27FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2×2+ω
φ(2)+φ(2)+φ(1)
ψ(2)+ψ(2)+ψ(1)
{n,n,1,2,3}
28FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2×3
φ(2)+φ(2)+φ(2)
ψ(2)+ψ(2)+ψ(2)
{n,n,1,1,4}
29FGH
FGH Veblen φ
FGH ψ-function
array notation
ω2×4
φ(2)+φ(2)+φ(2)+φ(2)
ψ(2)+ψ(2)+ψ(2)+ψ(2)
{n,n,1,1,5}
30proof theory
FGH

FGH Veblen φ
FGH ψ-function
array notation
EFA
ω3
φ(3) = φ(2)+φ(2)+φ(2)+...
ψ(3) = ψ(2)+ψ(2)+ψ(2)+...
{n,n,1,1,1,2} = {n,n,n,n,n}
31FGH
FGH Veblen φ
FGH ψ-function
array notation
ω3+1
φ(3)+φ(0)
ψ(3)+ψ(0)
{n,n,2,1,1,2}
32FGH
FGH Veblen φ
FGH ψ-function
array notation
ω3
φ(3)+φ(1)
ψ(3)+ψ(1)
{n,n,1,2,1,2}
33FGH
FGH Veblen φ
FGH ψ-function
array notation
ω32
φ(3)+φ(2)
ψ(3)+ψ(2)
{n,n,1,1,2,2}
34FGH
FGH Veblen φ
FGH ψ-function
array notation
ω3×2
φ(3)+φ(3)
ψ(3)+ψ(3)
{n,n,1,1,1,3} = {n,n,n,n,n,2}
35FGH
FGH Veblen φ
FGH ψ-function
array notation
ω3×3
φ(3)+φ(3)+φ(3)
ψ(3)+ψ(3)+ψ(3)
{n,n,2,1,1,4}
36FGH
FGH Veblen φ
FGH ψ-function
array notation
ω4
φ(4) = φ(3)+φ(3)+φ(3)+...
ψ(4) = ψ(3)+ψ(3)+ψ(3)+...
{n,n,1,1,1,1,2} = {n,n,n,n,n,n}
37FGH
FGH Veblen φ
FGH ψ-function
array notation
ω5
φ(5) = φ(4)+φ(4)+φ(4)+...
ψ(5) = ψ(4)+ψ(4)+ψ(4)+...
{n,n,1,1,1,1,1,2} = {n,n,n,n,n,n,n}
38proof theory
FGH

FGH Veblen φ
FGH ψ-function
array notation
BEAF
PRA, RCA0
ωω = ω↑↑2
φ(ω) = φ(φ(1)) = φ(φ(0)+φ(0)+φ(0)+...)
ψ(ω) = ψ(ψ(1)) = ψ(ψ(0)+ψ(0)+ψ(0)+...)
{n,n+2(2)2} = {n,} ~n+2 repeat
{n+2}&n
39FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω+1
φ(φ(1))+φ(0)
ψ(ψ(1))+ψ(0)
{n,n,2(2)2}
40FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω+2
φ(φ(1))+φ(0)+φ(0)
ψ(ψ(1))+ψ(0)+ψ(0)
{n,n,3(2)2}
41FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω
φ(φ(1))+φ(1)
ψ(ψ(1))+ψ(1)
{n,n,1,2(2)2}
42FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω2
φ(φ(1))+φ(2)
ψ(ψ(1))+ψ(2)
{n,n,1,1,2(2)2}
43FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω3
φ(φ(1))+φ(3)
ψ(ψ(1))+ψ(3)
{n,n,1,1,1,2(2)2}
44FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×2
φ(φ(1))+φ(φ(1))
ψ(ψ(1))+ψ(ψ(1))
{n,n+2(2)3} = {n,(2)2} ~n+2 repeat
45FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×3
φ(φ(1))+φ(φ(1))+φ(φ(1))
ψ(ψ(1))+ψ(ψ(1))+ψ(ψ(1))
{n,n+2(2)4} = {n,(2)3} ~n+2 repeat
46FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω+1 = ωω×ω
φ(φ(1)+φ(0)) = φ(φ(1))+φ(φ(1))+φ(φ(1))+...
ψ(ψ(1)+ψ(0)) = ψ(ψ(1))+ψ(ψ(1))+ψ(ψ(1))+...
{n,n+2(2)1,2} = {n,(2)n} ~n+2 repeat
47FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω+2 = ωω×ω×ω
φ(φ(1)+φ(0)+φ(0))
ψ(ψ(1)+ψ(0)+ψ(0))
{n,n+2(2)1,1,2} = {n,(2)n,n} ~n+2 repeat
48FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×2 = ωω×ωω
φ(φ(1)+φ(1))
ψ(ψ(1)+ψ(1))
{n,n+2(2)1(2)2} = {n,(2)n,} ~n+2 repeat first & ~n repeat second
49FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×2×2 = ωω×ωω×2
φ(φ(1)+φ(1))+φ(φ(1)+φ(1))
ψ(ψ(1)+ψ(1))+ψ(ψ(1)+ψ(1))
{n,n+2(2)1(2)3}
50FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×2+1 = ωω×ωω×ω
φ(φ(1)+φ(1)+φ(0)) = φ(φ(1)+φ(1))+φ(φ(1)+φ(1))+...
ψ(ψ(1)+ψ(1)+ψ(0)) = ψ(ψ(1)+ψ(1))+ψ(ψ(1)+ψ(1))+...
{n,n+2(2)1(2)1,2}
51FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×2+2 = ωω×ωω×ω×ω
φ(φ(1)+φ(1)+φ(0)+φ(0))
ψ(ψ(1)+ψ(1)+ψ(0)+ψ(0))
{n,n+2(2)1(2)1,1,2}
52FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×3 = ωω×ωω×ωω
φ(φ(1)+φ(1)+φ(1))
ψ(ψ(1)+ψ(1)+ψ(1))
{n,n+2(2)1(2)1(2)2}
53FGH
FGH Veblen φ
FGH ψ-function
array notation
ωω×4 = ωω×ωω×ωω×ωω
φ(φ(1)+φ(1)+φ(1)+φ(1))
ψ(ψ(1)+ψ(1)+ψ(1)+ψ(1))
{n,n+2(2)1(2)1(2)1(2)2}
54FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωω2 = ωω×ωω×ωω×ωω×...
φ(φ(2)) = φ(φ(1)+φ(1)+φ(1)+...)
ψ(ψ(2)) = ψ(ψ(1)+ψ(1)+ψ(1)+...)
{n,n(3)2} = {n,n+2(2)1(2)2} ~n-1 repeat
{n,2}&n
55FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωω3
φ(φ(3)) = φ(φ(2)+φ(2)+φ(2)+...)
ψ(ψ(3)) = ψ(ψ(2)+ψ(2)+ψ(2)+...)
{n,n(4)2}
{n,3}&n
56FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωω4
φ(φ(4)) = φ(φ(3)+φ(3)+φ(3)+...)
ψ(ψ(4)) = ψ(ψ(3)+ψ(3)+ψ(3)+...)
{n,n(5)2}
{n,4}&n
57FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω = ω↑↑3
φ(φ(ω)) = φ(φ(φ(1)))
ψ(ψ(ω)) = ψ(ψ(ψ(1)))
{n,n(n+1)2}={n,n(1,2)2}
{n,n}&n = {n,2,2}&n
58FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω+1
φ(φ(φ(1)))+φ(0)
ψ(ψ(ψ(1)))+ψ(0)
{n,n,2(1,2)2}
59FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω×2
φ(φ(φ(1)))+φ(φ(φ(1)))
ψ(ψ(ψ(1)))+ψ(ψ(ψ(1)))
{n,n(1,2)3}
60FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω+1 = ωωω×ω
φ(φ(φ(1))+φ(0)) = φ(φ(φ(1)))+φ(φ(φ(1)))+...
ψ(ψ(ψ(1))+ψ(0)) = ψ(ψ(ψ(1)))+ψ(ψ(ψ(1)))+...
{n,n(1,2)1,2}
61FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω
φ(φ(φ(1))+φ(1))
ψ(ψ(ψ(1))+ψ(1))
{n,n(1,2)1(2)2}
62FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω×2
φ(φ(φ(1))+φ(φ(1)))
ψ(ψ(ψ(1))+ψ(ψ(1)))
{n,n(1,2)1(1,2)2}
63FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωω×3
φ(φ(φ(1))+φ(φ(1))+φ(φ(1)))
ψ(ψ(ψ(1))+ψ(ψ(1))+ψ(ψ(1)))
{n,n(1,2)1(1,2)1(1,2)2}
64FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω+1 = ωωω×ω
φ(φ(φ(1)+φ(0))) = φ(φ(φ(1))+φ(φ(1))+...)
ψ(ψ(ψ(1)+ψ(0))) = ψ(ψ(ψ(1))+ψ(ψ(1))+...)
{n,n,(2,2)2} = {n,n(1,2)1(1,2)2} ~n-1 repeat
{n,n+1}&n
65FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω+2
φ(φ(φ(1)+φ(0)+φ(0)))
ψ(ψ(ψ(1)+ψ(0)+ψ(0)))
{n,n(3,2)2}
{n,n+2}&n
66FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω+3
φ(φ(φ(1)+φ(0)+φ(0)+φ(0)))
ψ(ψ(ψ(1)+ψ(0)+ψ(0)+ψ(0)))
{n,n(4,2)2}
{n,n+3}&n
67FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω×2
φ(φ(φ(1)+φ(1)))
ψ(ψ(ψ(1)+ψ(1)))
{n,n(1,3)2}
{n,n×2}&n
68FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω×3
φ(φ(φ(1)+φ(1)+φ(1)))
ψ(ψ(ψ(1)+ψ(1)+ψ(1)))
{n,n(1,4)2}
{n,n×3}&n
69FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω2
φ(φ(φ(2))) = φ(φ(φ(1)+φ(1)+φ(1)+...))
ψ(ψ(ψ(2))) = ψ(ψ(ψ(1)+ψ(1)+ψ(1)+...))
{n,n(1,1,2)2}
{n,{n,2}}&n
70FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω3
φ(φ(φ(3))) = φ(φ(φ(2)+φ(2)+φ(2)+...))
ψ(ψ(ψ(3))) = ψ(ψ(ψ(2)+ψ(2)+ψ(2)+...))
{n,n(1,1,1,2)2}
{n,{n,3}}&n
71FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωω4
φ(φ(φ(4))) = φ(φ(φ(3)+φ(3)+φ(3)+...))
ψ(ψ(ψ(4))) = ψ(ψ(ψ(3)+ψ(3)+ψ(3)+...))
{n,n(1,1,1,1,2)2}
{n,{n,4}}&n
72FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωω = ω↑↑4
φ(φ(φ(ω))) = φ(φ(φ(φ(1))))
ψ(ψ(ψ(ω))) = ψ(ψ(ψ(ψ(1))))
{n,n(1(2)2)2} = {n,n(1,2)2} ~n repeat
{n,{n,n}}&n ={n,3,2}&n
73FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωωω×2
φ(φ(φ(φ(1))))+φ(φ(φ(φ(1))))
ψ(ψ(ψ(ψ(1))))+ψ(ψ(ψ(ψ(1))))
{n,n(1(2)2)3}
74FGH
FGH Veblen φ
FGH ψ-function
array notation
ωωωω×2
φ(φ(φ(φ(1)))+φ(φ(φ(1))))
ψ(ψ(ψ(ψ(1)))+ψ(ψ(ψ(1))))
{n,n(2(2)2)2}
75FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωω×2
φ(φ(φ(φ(1))+φ(φ(1))))
ψ(ψ(ψ(ψ(1))+ψ(ψ(1))))
{n,n(1(2)3)2}
{n,{n×2,n}}&n
76FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωω×2
φ(φ(φ(φ(1)+φ(1))))
ψ(ψ(ψ(ψ(1)+ψ(1))))
{n,n(1(2)1(2)2)2}
{n,{n,n×2}}&n
77FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωω2
φ(φ(φ(φ(2)))) = φ(φ(φ(φ(1)+φ(1)+φ(1)+...)))
ψ(ψ(ψ(ψ(2)))) = ψ(ψ(ψ(ψ(1)+ψ(1)+ψ(1)+...)))
{n,n(1(3)2)2}
{n,{n,{n,2}}}&n
78FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωω3
φ(φ(φ(φ(3)))) = φ(φ(φ(φ(2)+φ(2)+φ(2)+...)))
ψ(ψ(ψ(ψ(3)))) = ψ(ψ(ψ(ψ(2)+ψ(2)+ψ(2)+...)))
{n,n(1(4)2)2}
{n,{n,{n,3}}}&n
79FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ωωωωω= ω↑↑5
φ(φ(φ(φ(ω)))) = φ(φ(φ(φ(φ(1)))))
ψ(ψ(ψ(ψ(ω)))) = ψ(ψ(ψ(ψ(ψ(1)))))
{n,n(1(1,2)2)2}
{n,{n,{n,n}}}&n = {n,4,2}&n
80FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ω↑↑6
φ(φ(φ(φ(φ(φ(1))))))
ψ(ψ(ψ(ψ(ψ(ψ(1))))))
{n,n(1(1(2)2)2)2}
{n,5,2}&n
81FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ω↑↑7
φ(φ(φ(φ(φ(φ(φ(1)))))))
ψ(ψ(ψ(ψ(ψ(ψ(ψ(1)))))))
{n,n(1(1(1,2)2)2)2}
{n,6,2}&n
82FGH
FGH Veblen φ
FGH ψ-function
array notation
BEAF
ω↑↑8
φ(φ(φ(φ(φ(φ(φ(φ(1))))))))
ψ(ψ(ψ(ψ(ψ(ψ(ψ(ψ(1))))))))
{n,n(1(1(1(2)2)2)2)2}
{n,7,2}&n
83name
proof theory
TON
FGH

FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
FGH ψ(cardinal)
FGH ψ(higher)

array notation (even)
array notation (odd)

SAN
BAN
BEAF
Small Cantor ordinal
PA, ACA0
limit of C0(a,b) - system
ωωωω... = ω↑↑ω = ε0 = 1Ɛ0
φ(1,0) = φ(φ(φ(φ(φ(φ(φ(φ(...)))))))) = φ(11)
ψ(Ω) = ψ(ψ(ψ(ψ(ψ(ψ(ψ(ψ(...)))))))) = 
ψ01(0))
ψ(1st uncountable)
ψ(Ф(1)) = ψ(I(0,0)) = ψ(χ(0))
{n,n(1...(2)...2)2} ~n/2-1 insert
{n,n(1...(1,2)...2)2} ~(n-1)/2-1 insert
{n,n(1.)2} = {n,n(1'2)2} = {n,n(1,,2)2}
{n,n((1))2} = {n,n(1\2)2} = {n,n(1/2)2}
{n,n,2}&n
84FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0+1
φ(1,0)+φ(0)
ψ(Ω)+ψ(0)
ψ01(0))+ψ0(0)
{n,n(2.)2} = {n,n(2'2)2}
{n,n((2))2} = {n,n(2\2)2} = {n,n(2/2)2}
85FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0+2
φ(1,0)+φ(0)+φ(0)
ψ(Ω)+ψ(0)+ψ(0)
ψ01(0))+ψ0(0)+ψ0(0)
{n,n(3.)2} = {n,n(3'2)2}
{n,n((3))2} = {n,n(3\2)2} = {n,n(3/2)2}
86FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0
φ(1,0)+φ(1)
ψ(Ω)+ψ(1)
ψ01(0))+ψ0(1)
{n,n(1,2.)2} = {n,n(1,2'2)2}
{n,n((1,2))2} = {n,n(1,2\2)2} = {n,n(1,2/2)2}
87FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε02
φ(1,0)+φ(2)
ψ(Ω)+ψ(2)
ψ01(0))+ψ0(2)
{n,n(1,1,2.)2} = {n,n(1,2'2)2}
{n,n((1,1,2))2} = {n,n(1,2\2)2} = {n,n(1,2/2)2}
88FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0ω
φ(1,0)+φ(φ(1))
ψ(Ω)+ψ(ψ(1))
ψ01(0))+ψ00(1))
{n,n(1(2)2.)2} = {n,n(1(2)2'2)2}
{n,n((1(2)2))2} = {n,n(1(2)2\2)2} = {n,n(1(2)2/2)2}
89FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0ωω
φ(1,0)+φ(φ(φ(1)))
ψ(Ω)+ψ(ψ(ψ(1)))
ψ01(0))+ψ000(1)))
{n,n(1(1,2)2.)2} = {n,n(1(1,2)2'2)2}
{n,n((1(1,2)2))2} = {n,n(1(1,2)2\2)2} = {n,n(1(1,2)2/2)2}
90FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN (even)
SAN (odd)

BAN (even)
BAN (odd)
ε0×2
φ(1,0)+φ(φ(φ(...))) = φ(1,0)+φ(1,0)
ψ(Ω)+ψ(ψ(ψ(...))) = ψ(Ω)+ψ(Ω)
ψ01(0))+ψ01(0))
{n,n(1.2.)2} = {n,n(1(1'2)2'2)2} = {n,n(1(...2...)2.)2} ~n/2 insert
{n,n(1.2.)2} = {n,n(1(1'2)2'2)2} = {n,n(1(...1,2...)2.)2} ~(n-1)/2 insert
{n,n((1((1))2))2} = {n,n(1(1\2)2\2)2} = {n,n(1(1/2)2/2)2} = {n,n((1...(2)...2))2}~n/2 insert
{n,n((1((1))2))2} = {n,n(1(1\2)2\2)2} = {n,n(1(1/2)2/2)2} = {n,n((1...(1,2)...2))2}~(n-1)/2 insert
91FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0×3
φ(1,0)+φ(1,0)+φ(1,0)
ψ(Ω)+ψ(Ω)+ψ(Ω)
ψ01(0))+ψ01(0))+ψ01(0))
{n,n(1.3.)2} = {n,n(1(1'2)3'2)2}
{n,n((1((1))3))2} = {n,n(1(1\2)3\2)2} = {n,n(1(1/2)3/2)2}
92FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0×ω = ωε0+1
φ(φ(1,0)+1)  = φ(1,0)+φ(1,0)+...
ψ(Ω+ψ(0)) = ψ(Ω+1) = ψ(Ω)+ψ(Ω)+...
ψ01(0)+ψ0(0))
{n,n(1.1,2.)2} = {n,n(1(1'2)1,2'2)2}
{n,n((1((1))1,2))2} = {n,n(1(1\2)1,2\2)2} = {n,n(1(1/2)1,2/2)2}
93FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0×ω×ω = ωε0+2
φ(φ(1,0)+2)
ψ(Ω+ψ(0)+ψ(0)) = ψ(Ω+2)
ψ01(0)+ψ0(0)+ψ0(0))
{n,n(1.1(2)2.)2} = {n,n(1(1'2)1(2)2'2)2}
{n,n((1((1))1,1,2))2} = {n,n(1(1\2)1(2)2\2)2} = {n,n(1(1/2)1(2)2/2)2}
94FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε0×ω×ω×ω×... = ε0×ωω = ωε0
φ(φ(1,0)+φ(1))
ψ(Ω+ψ(1)) = ψ(Ω+ω)
ψ01(0)+ψ0(1))
{n,n(1.1,1,2.)2} = {n,n(1(1'2)1,1,2'2)2}
{n,n((1((1))1,1,2))2} = {n,n(1(1\2)1,1,2\2)2} = {n,n(1(1/2)1,1,2/2)2}
95FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε02 = ωε0×2
φ(φ(1,0)+φ(1,0))
ψ(Ω+ψ(Ω))
ψ01(0)+ψ01(0)))
{n,n(1.1.2.)2} = {n,n(1(1'2)1(1'2)2'2)2}
{n,n((1((1))1((1))2))2} = {n,n(1(1\2)1(1\2)2\2)2} = {n,n(1(1/2)1(1/2)2/2)2}
96FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ε03 = ωε0×3
φ(φ(1,0)+φ(1,0)+φ(1,0))
ψ(Ω+ψ(Ω)+ψ(Ω))
ψ01(0)+ψ01(0))+ψ01(0)))
{n,n(1.1.1.2.)2} = {n,n(1(1'2)1(1'2)1(1'2)2'2)2}
{n,n((1((1))1((1))1((1))2))2} = {n,n(1(1\2)1(1\2)1(1\2)2\2)2} = {n,n(1(1/2)1(1/2)1(1/2)2/2)2}
97FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε0ω = ωε0×ω = ωωε0+1
φ(φ(φ(1,0)+1))
ψ(Ω+ψ(Ω+ψ(0))) = ψ(Ω+ψ(Ω)+ψ(Ω)+...)
ψ01(0)+ψ01(0)+ψ0(0)))
{n,n(1(2.)2.)2} = {n,n(1(2'2)2'2)2} = {n,n(1.2.)2} ~n repeat = {n,n(1(1'2)2'2)2} ~n repeat
{n,n((1((2))2))2} = {n,n(1(2\2)2\2)2} = {n,n(1(2/2)2/2)2} = {n,n((1((1))2))2} ~n repeat
{n×{n,n,2}}&n
98FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε0ε0ω = ωωε0×ω = ωωωε0+1
φ(φ(φ(φ(1,0)+1)))
ψ(Ω+ψ(Ω+ψ(Ω+ψ(0))))
ψ01(0)+ψ01(0)+ψ01(0)+ψ0(0))))
{n,n(1(1.1,2.)2.)2} = {n,n(1(1(1'2)1,2'2)2'2)2}
{n,n((1((1((1))1,2))2))2} = {n,n(1(1(1\2)1,2\2)2\2)2} = {n,n(1(1(1/2)1,2/2)2/2)2}
{{n,n,2},{{n,n,2},n}}&n
99FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε0ε0ε0ω = ωωωε0×ω = ωωωωε0+1
φ(φ(φ(φ(φ(1,0)+1))))
ψ(Ω+ψ(Ω+ψ(Ω+ψ(Ω+ψ(0)))))
ψ01(0)+ψ01(0)+ψ01(0)+ψ01(0)+ψ0(0)))))
{n,n(1(1(2.)2.)2.)2} = {n,n(1(1(2'2)2'2)2'2)2}
{n,n((1((1((2))2))2))2} = {n,n(1(1(2'2)2'2)2'2)2} = {n,n(1(1(2'2)2'2)2'2)2}
{{n,n,2},{{n,n,2},{{n,n,2},n}}}&n
100FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε0ε0ε0ε0ω = ωωωωε0×ω = ωωωωωε0+1
φ(φ(φ(φ(φ(φ(1,0)+1)))))
ψ(Ω+ψ(Ω+ψ(Ω+ψ(Ω+ψ(Ω+ψ(0))))))
ψ01(0)+ψ01(0)+ψ01(0)+ψ01(0)+ψ01(0)+ψ0(0))))))
{n,n(1(1(1(1.1,2.)2.)2.)2.)2} = {n,n(1(1(1(1'2)1,2'2)2'2)2'2)2}
{n,n((1((1((1((1))1,2))2))2))2} = {n,n(1(1(2'2)2'2)2'2)2} = {n,n(1(1(2'2)2'2)2'2)2}
{{n,n,2},{{n,n,2},{{n,n,2},{{n,n,2},n}}}}&n
101FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN (even)
SAN (odd)

BAN (even)
BAN (odd)
BEAF
ε1 = ε0ε0ε0ε0...= ωωω...ε0+1 = 1Ɛ1
φ(1,1) = φ(φ(φ(φ(...φ(1,0)+1...)))) = φ(1110)
ψ(Ω×2) = ψ(Ω+Ω) = ψ(Ω+ψ(Ω+ψ(Ω+ψ(...))))
ψ01(0)+ψ1(0))
{n,n(1.2)2} = {n,n(1'3)2} = {n,n(1...(2.)...2.)2} ~n/2 insert
{n,n(1.2)2} = {n,n(1'3)2} = {n,n(1...(1.1,2.)...2.)2} ~(n-1)/2 insert
{n,n(((1)))2} = {n,n(1\3)2} = {n,n(1/3)2} = {n,n((1...((2))...2))2} ~n/2 insert
{n,n(((1)))2} = {n,n(1\3)2} = {n,n(1/3)2} = {n,n((1...((1((1))1,2))...2))2} ~(n-1)/2 insert
{n,n×2,2}&n
102FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε2 = ε1ε1ε1ε1...= ωωω...ε1+1= 1Ɛ2
φ(1,2) = φ(φ(φ(φ(...φ(1,1)+1...)))) = φ(1120)
ψ(Ω×3)
ψ01(0)+ψ1(0)+ψ1(0))
{n,n(1.3)2} = {n,n(1'4)2}
{n,n((((1))))2} = {n,n(1\4)2} = {n,n(1/4)2}
{n,n×3,2}&n
103FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ε3 = ε2ε2ε2ε2...= ωωω...ε2+1= 1Ɛ3
φ(1,3) = φ(φ(φ(φ(...φ(1,2)+1...)))) = φ(1130)
ψ(Ω×4)
ψ01(0)+ψ1(0)+ψ1(0)+ψ1(0))
{n,n(1.4)2} = {n,n(1'5)2}
{n,n(((((1)))))2} = {n,n(1\5)2} = {n,n(1/5)2}
{n,n×4,2}&n
104FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εω = 1Ɛω
φ(1,ω) = φ(1,φ(1)) = φ(11ω0)
ψ(Ω×ω)
ψ01(1))
{n,n(1.n+1)2} = {n,n(1'1,2)2}
{n,n(1\1,2)2} = {n,n(1/1,2)2} = {n,n(...(1)...)2} ~n+2 insert
{n,{n,2},2}&n
105FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εω×ω = ωεω+1
φ(φ(1,φ(1))+1)
ψ(Ω×ω+1)
ψ01(1)+ψ0(0))
{n,n(1(1'1,2)1,2'1,2)2}
{n,n(1(1\2)1,2\1,2)2} = {n,n(1(1/2)1,2/1,2)2}
106FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωωεω+1
φ(φ(φ(1,φ(1))+1))
ψ(Ω×ω+ψ(Ω×ω+1))
ψ01(1)+ψ01(1)+ψ0(0)))
{n,n(1(2'1,2)2'1,2)2}
{n,n(1(2\2)2\1,2)2} = {n,n(1(2/2)2/1,2)2}
107FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωωωεω+1
φ(φ(φ(φ(1,φ(1))+1)))
ψ(Ω×ω+ψ(Ω×ω+ ψ(Ω×ω+1)))
ψ01(1)+ψ01(1)+ψ01(1)+ψ0(0))))
{n,n(1(1(1'1,2)1,2'1,2)2'1,2)2}
{n,n(1(1(1\2)1,2\2)2\1,2)2} = {n,n(1(1(1/2)1,2/2)2/1,2)2}
108FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωωωωεω+1
φ(φ(φ(φ(φ(1,φ(1))+1))))
ψ(Ω×ω+ψ(Ω×ω+ψ(Ω×ω+ψ(Ω×ω+1))))
ψ01(1)+ψ01(1)+ψ01(1)+ψ01(1)+ψ0(0)))))
{n,n(1(1(2'1,2)2'1,2)2'1,2)2}
{n,n(1(1(2\2)2\2)2\1,2)2} = {n,n(1(1(2/2)2/2)2/1,2)2}
109FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εω+1 = ωωω...εω+1
φ(φ(φ(φ(...φ(1,φ(1))+1...)))) = φ(1,φ(1)+1)
ψ(Ω×ω+Ω) = ψ(Ω×(ω+1))
ψ01(1)+ψ1(0))
{n,n(1'2,2)2}
{n,n(1\2,2)2} = {n,n(1/2,2)2}
{n,{n,2}+n,2}&n
110FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εω+2 = ωωω...εω+1+1
φ(φ(φ(φ(...φ(1,φ(1)+1)+1...)))) = φ(1,φ(1)+2)
ψ(Ω×ω+Ω+Ω) = ψ(Ω×(ω+2))
ψ01(1)+ψ1(0)+ψ1(0))
{n,n(1'3,2)2}
{n,n(1\3,2)2} = {n,n(1/3,2)2}
{n,{n,2}+n×2,2}&n
111FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εω×2
φ(1,φ(1)+φ(1))
ψ(Ω×ω+Ω×ω) = ψ(Ω×ω×2)
ψ01(1)+ψ1(1)) = ψ01(1)+ψ1(0)+ψ1(0)+...)
{n,n(1'1,3)2}
{n,n(1\1,3)2} = {n,n(1/1,3)2}
{n,{n×2,2},2}&n
112FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εω2
φ(1,φ(2))
ψ(Ω×ω×ω) = ψ(Ω×ω2)
ψ01(2))
{n,n(1'1,1,2)2}
{n,n(1\1,1,2)2} = {n,n(1/1,1,2)2}
{n,{n,3},2}&n
113FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εωω
φ(1,φ(φ(1)))
ψ(Ω×ωω)
ψ010(1)))
{n,n(1'1(2)2)2}
{n,n(1\1(2)2)2} = {n,n(1/1(2)2)2}
{n,{n,n},2}&n
114FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εωωω
φ(1,φ(φ(φ(1))))
ψ(Ω×ωωω)
ψ0100(1))))
{n,n(1'1(1,2)2)2}
{n,n(1\1(1,2)2)2} = {n,n(1/1(1,2)2)2}
{n,{n,{n,n}},2}&n
115FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εωωωω
φ(1,φ(φ(φ(φ(1)))))
ψ(Ω×ωωωω)
ψ01000(1)))))
{n,n(1'1(1(2)2)2)2}
{n,n(1\1(1(2)2)2)2} = {n,n(1/1(1(2)2)2)2}
{n,{n,{n,{n,n}}},2}&n
116FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εε0 = εωωωω...
φ(1,ε0) = φ(1,φ(1,0))
ψ(Ω×ψ(Ω)) = ψ(Ω×ωωω...)
ψ0101(0))))
{n,n(1'1.2)2} = {n,n(1'1(1'2)2)2}
{n,n(1\1(1\3)2)2} = {n,n(1/1(1/3)2)2}
{n,{n,n,2},2}&n
117FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εε1
φ(1,ε1) = φ(1,φ(1,1))
ψ(Ω×ψ(Ω×2))
ψ0101(0)+ψ1(0))))
{n,n(1'1(1'3)2)2}
{n,n(1\1(1\2)2)2} = {n,n(1/1(1/2)2)2}
{n,{n,n×2,2},2}&n
118FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εεω
φ(1,εω) = φ(1,φ(1,φ(1)))
ψ(Ω×ψ(Ω×ω))
ψ0101(1))))
{n,n(1'1(1'1,2)2)2}
{n,n(1\1(1\1,2)2)2} = {n,n(1/1(1/1,2)2)2}
{n,{n,{n,2},2},2}&n
119FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εεε0
φ(1,εε0) = φ(1,φ(1,φ(1,0)))
ψ(Ω×ψ(Ω×ψ(Ω)))
ψ010101(0))))))
{n,n(1'1(1'1.2)2)2}
{n,n(1\1(1\1(1\2)2)2)2} = {n,n(1/1(1/1(1/2)2)2)2}
{n,{n,{n,n,2},2},2}&n
120FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
εεεε0
φ(1,εεε0) = φ(1,φ(1,φ(1,φ(1,0))))
ψ(Ω×ψ(Ω×ψ(Ω×ψ(Ω))))
ψ01010101(0)))))))
{n,n(1'1(1'1(1'1.2)2)2)2}
{n,n(1\1(1\1(1\1(1\2)2)2)2)2} = {n,n(1/1(1/1(1/1(1/2)2)2)2)2}
{n,{n,{n,{n,n,2},2},2},2}&n
121name
FGH

FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Cantor ordinal
ζ0 = εεεε... = 2Ɛ0
φ(2,0) = φ(1,φ(1,φ(1,φ(1,...))) = φ(21)
ψ(Ω2) = ψ(Ω×Ω) = ψ(Ω×ψ(Ω×ψ(Ω×ψ(...))))
ψ011(0)))
{n,n(1'1'2)2} = {n,n(1'1...(1'1.2)...2)2} ~n+1 insert
{n,n(1\1\2)2} = {n,n(1/1/2)2} = {n,n(1\1...(1\2)...2)2} ~n insert  = {n,n(1/1...(1/2)...2)2} ~n insert
{n,n,3}&n = {n,...n,...2}&n ~n insert
122FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζ0+1
φ(2,0)+1
ψ(Ω2)+ψ(0)
ψ011(0)))+ψ0(0)
{n,n(2'1'2)2}
{n,n(2\1\2)2} = {n,n(2/1/2)2}
123FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζ0×ω = ωζ0+1
φ(φ(2,0)+1)
ψ(Ω2+ψ(0))
ψ011(0))+ψ0(0))
{n,n(1(1'1'2)1,2'1'2)2}
{n,n(1(1\1\2)1,2\1\2)2} = {n,n(1(1/1/2)1,2/1/2)2}
124FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εζ0+1 = ζ0ζ0ζ0ζ0...= ωωω...ζ0+1
φ(1,φ(2,0)+1)
ψ(Ω2+Ω)
ψ011(0))+ψ1(0))
{n,n(1'2'2)2}
{n,n(1\2\2)2} = {n,n(1/2/2)2}
125FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εζ0+2 = ωωω...εζ0+1
φ(1,φ(2,0)+2)
ψ(Ω2+Ω+Ω)
ψ011(0))+ψ1(0)+ψ1(0))
{n,n(1'3'2)2}
{n,n(1\3\2)2} = {n,n(1/3/2)2}
126FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εεζ0+1 = εζ0ζ0ζ0ζ0...
φ(1,φ(1,φ(2,0)+1))
ψ(Ω2+Ω×ψ(Ω2+Ω))
ψ011(0))+ψ1011(0))+ψ1(0)))))
{n,n(1'1(1'2'2)2'2)2}
{n,n(1\1(1\2\2)2\2)2} = {n,n(1/1(1/2/2)2/2)2}
127FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εεεζ0+1= εεζ0ζ0ζ0ζ0...
φ(1,φ(1,φ(1,φ(2,0)+1)))
ψ(Ω2+Ω×ψ(Ω2+Ω×ψ(Ω2+Ω)))
ψ011(0))+ψ1011(0))+ψ1011(0))+ψ1(0))))))))
{n,n(1'1(1'1(1'2'2)2'2)2'2)2}
{n,n(1\1(1\1(1\2\2)2\2)2\2)2} = {n,n(1/1(1/1(1/2/2)2/2)2/2)2}
128FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζ1 = εεε...ζ0+1= 2Ɛ1
φ(2,1) = φ(1,φ(1,φ(1,φ(1,...φ(2,0)+1...)))) = φ(2110)
ψ(Ω2×2) = ψ(Ω2+Ω×Ω) = ψ(Ω2+Ω×ψ(Ω2+Ω×ψ(Ω2+Ω×ψ(Ω2+Ω×...))))
ψ011(0))+ψ11(0)))
{n,n(1'1'3)2} = {n,n(1'1...(1'2'2)...2'2)2} ~n-1 insert
{n,n(1\1\3)2} = {n,n(1\1...(1\2\2)...2\2)2} ~n-1 insert = {n,n(1/1/3)2} = {n,n(1/1...(1/2/2)...2/2)2} ~n-1 insert
{n,n×2,3}&n
129FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζ2 = εεε...ζ1+1= 2Ɛ2
φ(2,2) = φ(1,φ(1,φ(1,φ(1,...φ(2,1)+1...))) = φ(2120)
ψ(Ω2×3)
ψ011(0))+ψ11(0))+ψ11(0)))
{n,n(1'1'4)2}
{n,n(1\1\4)2} = {n,n(1/1/4)2}
{n,n×3,3}&n
130FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζω = 2Ɛω
φ(2,ω) = φ(2,φ(1)) = φ(21ω0)
ψ(Ω2×ψ(1))
ψ011(0)+ψ0(0)))
{n,n(1'1'1,2)2}
{n,n(1\1\1,2)2} = {n,n(1/1/1,2)2}
{n,{n,2},3}&n
131FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζε0
φ(2,φ(1,0))
ψ(Ω2×ψ(Ω))
ψ011(0)+ψ01(0))))
{n,n(1'1'1.2)2}
{n,n(1\1\1(1\2)2)2} = {n,n(1/1/1(1/2)2)2}
{n,{n,n,2},3}&n
132FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζζ0
φ(2,φ(2,0))
ψ(Ω2×ψ(Ω2))
ψ011(0)+ψ011(0)))))
{n,n(1'1'1(1'1'2)2)2}
{n,n(1\1\1(1\1\2)2)2} = {n,n(1/1/1(1/1/2)2)2}
{n,{n,n,3},3}&n
133FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ζζζ0
φ(2 φ(2,φ(2,0)))
ψ(Ω2×ψ(Ω2×ψ(Ω2)))
ψ011(0)+ψ011(0)+ψ011(0)))))))
{n,n(1'1'1(1'1'1(1'1'2)2)2)2}
{n,n(1\1\1(1\1\1(1\1\2)2)2)2} = {n,n(1/1/1(1/1/1(1/1/2)2)2)2}
{n,{n,{n,n,3},3},3}&n
134name
FGH

FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Large Cantor ordinal
η0 = ζζζζ... = 3Ɛ0
φ(3,0) = φ(2,φ(2,φ(2,φ(2,...))) = φ(31)
ψ(Ω3) = ψ(Ω2×Ω)= ψ(Ω2×ψ(Ω2×ψ(Ω2×ψ(...))))
ψ011(0)+ψ1(0)))
{n,n(1'1'1'2)2} = {n,n(1'1'1...(1'1'2)...2)2} ~n-1 insert
{n,n(1\1\1\2)2} = {n,n(1\1\1...(1\1\2)...2)2} ~n-1 insert = {n,n(1/1/1/2)2} = {n,n(1/1/1...(1/1/2)...2)2} ~n-1 insert
{n,n,4}&n
135FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
η0+1
φ(3,0)+1
ψ(Ω3)+ψ(0)
ψ011(0)+ψ1(0)))+ψ0(0)
{n,n(2'1'1'2)2}
{n,n(2\1\1\2)2} = {n,n(2/1/1/2)2}
136FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
η0×ω = ωη0+1
φ(φ(3,0)+1)
ψ(Ω3+ψ(0))
ψ011(0)+ψ1(0))+ψ0(0))
{n,n(1(1'1'1'2)1,2'1'1'2)2}
{n,n(1(1\1\1\2)1,2\1\1\2)2} = {n,n(1(1/1/1/2)1,2/1/1/2)2}
137FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εη0+1 = η0η0η0η0... = ωωω...η0+1
φ(1,φ(3,0)+1)
ψ(Ω3+Ω)
ψ011(0)+ψ1(0))+ψ1(0))
{n,n(1'2'1'2)2}
{n,n(1\2\1\2)2} = {n,n(1/2/1/2)2}
138FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εη0+2 = ωωω...εη0+1
φ(1,φ(3,0)+2)
ψ(Ω3+Ω+Ω)
ψ011(0)+ψ1(0))+ψ1(0)+ψ1(0))
{n,n(1'3'1'2)2}
{n,n(1\3\1\2)2} = {n,n(1/3/1/2)2}
139FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζη0+1 = εεε...η0+1
φ(2,φ(3,0)+1)
ψ(Ω32)
ψ011(0)+ψ1(0))+ψ11(0)))
{n,n(1'1'2'2)2}
{n,n(1\1\2\2)2} = {n,n(1/1/2/2)2}
140FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζη0+2 = εεε...εη0+1
φ(2,φ(3,0)+2)
ψ(Ω322)
ψ011(0)+ψ1(0))+ψ11(0))+ψ11(0)))
{n,n(1'1'3'2)2}
{n,n(1\1\3\2)2} = {n,n(1/1/3/2)2}
141FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζζη0+1 = ζεε...η0+1
φ(2,φ(2,φ(3,0)+1))
ψ(Ω32×ψ(Ω32))
ψ011(0)+ψ1(0))+ψ11011(0)+ψ1(0))+ψ11(0)))))
{n,n(1'1'1(1'1'2'2)2'2)2}
{n,n(1\1\1(1\1\2\2)2\2)2} = {n,n(1/1/1(1/1/2/2)2/2)2}
142FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
η1 = ζζζ...η0+1= 3Ɛ1
φ(3,1) = φ(2,φ(2,φ(2,φ(2,...φ(3,0)+1...)))) = φ(3110)
ψ(Ω3×2) = ψ(Ω3+ψ(Ω3+ψ(Ω3+ψ(...))))
ψ011(0)+ψ1(0))+ψ11(0)+ψ1(0)))
{n,n(1'1'1'3)2}
{n,n(1\1\1\3)2} = {n,n(1/1/1/3)2}
{n,n×2,4}&n
143FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
η2 = ζζζ...η1+1 = 3Ɛ2
φ(3,2) = φ(2,φ(2,φ(2,φ(2,...φ(3,1)+1...))) = φ(3120)
ψ(Ω3×3)
ψ011(0)+ψ1(0))+ψ11(0)+ψ1(0))+ψ11(0)+ψ1(0)))
{n,n(1'1'1'4)2}
{n,n(1\1\1\4)2} = {n,n(1/1/1/4)2}
{n,n×3,4}&n
144FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ηω = 3Ɛω
φ(3,ω) = φ(3,φ(1)) = φ(31ω0)
ψ(Ω3×ψ(1))
ψ011(0)+ψ1(0)+ψ0(0)))
{n,n(1'1'1'1,2)2}
{n,n(1\1\1\1,2)2} = {n,n(1/1/1/1,2)2}
{n,{n,2},4}&n
145FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ηε0
φ(3,φ(1,0))
ψ(Ω3×ψ(Ω))
ψ011(0)+ψ1(0)+ψ01(0))))
{n,n(1'1'1'1(1'2)2)2}
{n,n(1\1\1\1(1\2)2)2} = {n,n(1/1/1/1(1/2)2)2}
{n,{n,n,2},4}&n
146FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ηζ0
φ(3,φ(2,0))
ψ(Ω3×ψ(Ω2))
ψ011(0)+ψ1(0)+ψ011(0)))))
{n,n(1'1'1'1(1'1'2)2)2}
{n,n(1\1\1\1(1\1\2)2)2} = {n,n(1/1/1/1(1/1/2)2)2}
{n,{n,n,3},4}&n
147FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ηη0
φ(3,φ(3,0))
ψ(Ω3×ψ(Ω3))
ψ011(0)+ψ1(0)+ψ011(0)+ψ1(0)))))
{n,n(1'1'1'1(1'1'1'2)2)2}
{n,n(1\1\1\1(1\1\1\2)2)2} = {n,n(1/1/1/1(1/1/1/2)2)2}
{n,{n,n,4},4}&n
148FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ηηηη... = 4Ɛ0
φ(4,0) = φ(3,φ(3,φ(3,φ(3,...))) = φ(41)
ψ(Ω4) = ψ(Ω3×ψ(Ω3×ψ(Ω3×ψ(...))))
ψ011(0)+ψ1(0)+ψ1(0)))
{n,n(1'1'1'1'2)2}
{n,n(1\1\1\1\2)2} = {n,n(1/1/1/1/2)2}
{n,n,5}&n
149FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
5Ɛ0 = 4Ɛ4Ɛ4Ɛ4Ɛ...
φ(5,0) = φ(4,φ(4,φ(4,φ(4,...))) = φ(51)
ψ(Ω5) = ψ(Ω4×ψ(Ω4×ψ(Ω4×ψ(...))))
ψ011(0)+ψ1(0)+ψ1(0)+ψ1(0)))
{n,n(1'1'1'1'1'2)2}
{n,n(1\1\1\1\1\2)2} = {n,n(1/1/1/1/1/2)2}
{n,n,6}&n
150FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ0
φ(ω,0) = φ(φ(1),0) = φ(ω1)
ψ(Ωω)
ψ011(1)))
{n,n(1(2')2)2}={n,n(1'2)2} ~n-1 repeat = {n,n(2,,2)2}
{n,n(1\\2)2}={n,n(1\2)2} ~n-1 repeat = {n,n(1(2\22)2)2} = {n,n(1//2)2} ={n,n(1/2)2} ~n-1 repeat = {n,n(1(2/22)2)2}
{n,n,n}&n
151FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
1ƐωƐ0+1
φ(1,φ(ω,0)+1)  = εφ(ω,0)+1
ψ(Ωω+Ω)
ψ011(1))+ψ1(0)))
{n,n(1'2(2')2)2}
{n,n(1\1\\2)2} = {n,n(1/1//2)2}
{{n,n,n},n,2}&n
152FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
2ƐωƐ0+1
φ(2,φ(ω,0)+1)  = ζφ(ω,0)+1
ψ(Ωω2)
ψ011(1))+ψ11(0))))
{n,n(1'1'2(2')2)2}
{n,n(1\1\1\\2)2} = {n,n(1/1/1//2)2}
{{n,n,n},n,3}&n
153FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
3ƐωƐ0+1
φ(3,φ(ω,0)+1)  = ηφ(ω,0)+1
ψ(Ωω3)
ψ011(1))+ψ11(0)+ψ1(0))))
{n,n(1'1'1'2(2')2)2}
{n,n(1\1\1\1\\2)2} = {n,n(1/1/1/1//2)2}
{{n,n,n},n,4}&n
154FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ1
φ(ω,1) = φ(φ(1),1)  = φ(ω110)
ψ(Ωω×2) = ψ(Ωωω)
ψ011(1))+ψ11(1))) = ψ011(1))+ψ11(0)+ψ1(0)+ψ1(0)+...)))
{n,n(1(2')3)2}  = {n,n(1'2(2')2)2} ~n repeat
{n,n(1\\3)2} = {n,n(1(2\22)3)2} ={n,n(1\1\\2)2} ~n repeat = {n,n(1//3)2} = {n,n(1(2/22)3)2} = {n,n(1/1//2)2} ~n repeat
{n,n×2,n}&n
155FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐω
φ(ω,ω) = φ(φ(1),φ(1)) = φ(ω1ω0)
ψ(Ωω×ω) = ψ(Ωω×ψ(1))
ψ011(1)+ψ0(0))) = ψ011(1))+ψ11(1))+ψ11(1))+...)
{n,n(1(2')1,2)2}
{n,n(1\\1,2)2} = {n,n(1//1,2)2}
{n,{n,2},n}&n
156FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ1Ɛ0
φ(ω,ε0) = φ(φ(1),φ(1,0))
ψ(Ωω×ε0) = ψ(Ωω×ψ(Ω))
ψ011(1)+ψ01(0))))
{n,n(1(2')1.2)2} = {n,n(1(2')1(1'2)2)2}
{n,n(1\\1(1\2)2)2} = {n,n(1//1(1/2)2)2}
{n,{n,n,2},n}&n
157FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ2Ɛ0
φ(ω,ζ0) = φ(φ(1),φ(2,0))
ψ(Ωω×ζ0) = ψ(Ωω×ψ(Ω2))
ψ011(1)+ψ011(0)))))
{n,n(1(2')1(1'1'2)2)2}
{n,n(1\\1(1\1\2)2)2} = {n,n(1//1(1/1/2)2)2}
{n,{n,n,3},n}&n
158FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ3Ɛ0
φ(ω,η0) = φ(φ(1),φ(3,0))
ψ(Ωω×η0) = ψ(Ωω×ψ(Ω3))
ψ011(1)+ψ011(0)+ψ1(0)))))
{n,n(1(2')1(1'1'1'2)2)2}
{n,n(1\\1(1\1\1\2)2)2} = {n,n(1//1(1/1/1/2)2)2}
{n,{n,n,4},n}&n
159FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐωƐ0
φ(ω,φ(ω,0)) = φ(φ(1),φ(φ(1),0))
ψ(Ωω×ψ(Ωω))
ψ011(1)+ψ011(1)))))
{n,n(1(2')1(1(2')2)2)2}
{n,n(1\\1(1\\2)2)2} = {n,n(1//1(1//2)2)2}
{n,{n,n,n},n}&n
160FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐωƐωƐ0
φ(ω,φ(ω,φ(ω,0))) = φ(φ(1),φ(φ(1),φ(φ(1),0)))
ψ(Ωω×ψ(Ωω×ψ(Ωω)))
ψ011(1)+ψ011(1)+ψ011(1)))))))
{n,n(1(2')1(1(2')1(1(2')2)2)2)2
{n,n(1\\1(1\\1(1\\2)2)2)2} = {n,n(1//1(1//1(1//2)2)2)2}
{n,{n,{n,n,n},n},n}&n
161FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω+1Ɛ0
φ(ω+1,0) = φ(ω,φ(ω,φ(ω,...))) = φ(φ(1),φ(φ(1),φ(φ(1),...)))
ψ(Ωω+1) = ψ(Ωω×ψ(Ωω×ψ(...)))
ψ011(1)+ψ1(0)))
{n,n(1(2')1'2)2} = {n,n(1(2')1(1(2')...2)2)2} ~n-1 insert
{n,n(1\\1\2)2} = {n,n(1\\1(1\\...2)2)2} ~n-1 insert = {n,n(1//1/2)2} = {n,n(1//1(1//...2)2)2}  ~n-1 insert
{n,...n,...n}&n ~n insert = {n,n,n+1}&n
162FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω+2Ɛ0
φ(ω+2,0)
ψ(Ωω+2)
ψ011(1)+ψ1(0)+ψ1(0)))
{n,n(1(2')1'1'2)2}
{n,n(1\\1\1\2)2} = {n,n(1//1/1/2)2}
{n,n,n+2}&n
163FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω+3Ɛ0
φ(ω+3,0)
ψ(Ωω+3)
ψ011(1)+ψ1(0)+ψ1(0)+ψ1(0)))
{n,n(1(2')1'1'1'2)2}
{n,n(1\\1\1\1\2)2} = {n,n(1//1/1/1/2)2}
{n,n,n+3}&n
164FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω×2Ɛ0
φ(ω×2,0)
ψ(Ωω×2)
ψ011(1)+ψ1(1)))
{n,n(1(2')1(2')2)2}
{n,n(1\\1\\2)2} = {n,n(1//1//2)2}
{n,n,n×2}&n
165FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω×3Ɛ0
φ(ω×3,0)
ψ(Ωω×3)
ψ011(1)+ψ1(1)+ψ1(1)))
{n,n(1(2')1(2')1(2')2)2}
{n,n(1\\1\\1\\2)2} = {n,n(1//1//1//2)2}
{n,n,n×3}&n
166FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω2Ɛ0
φ(ω2,0) = φ(φ(2),0)
ψ(Ωω2) = ψ011(2)))
ψ011(2)))
{n,n(1(3')2)2} = {n,n(1(2')...2)2} ~n repeat
{n,n(1\\\\2)2} = {n,n(1\\2)2} ~n repeat = {n,n(1(4\22)2)2} = {n,n(1////2)2} = {n,n(1//2)2} ~n repeat = {n,n(1(4/22)2)2}
{n,n,{n,2}}&n
167FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ω3Ɛ0
φ(ω3,0) = φ(φ(3),0)
ψ(Ωω3) = ψ011(3)))
ψ011(3)))
{n,n(1(4')2)2}
{n,n(1\\\\2)2} = {n,n(1(4\22)2)2} = {n,n(1////2)2} = {n,n(1(4/22)2)2}
{n,n,{n,3}}&n
168FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωωƐ0
φ(ωω,0) = φ(φ(φ(1)),0)
ψ(Ωωω) = ψ0110(1))))
ψ0110(1))))
{n,n(1(1,2')2)2}
{n,n(1\2)2} ~n+1 repeat = {n,n(1(1,2\22)2)2} = {n,n(1/2)2} ~n+1 repeat = {n,n(1(1,2/22)2)2}
{n,n,{n,n}}&n = {n,n,{n,2,2}}&n
169FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωωωƐ0
φ(ωωω,0) = φ(φ(φ(φ(1))),0)
ψ(Ωωωω) = ψ01100(1)))))
ψ01100(1)))))
{n,n(1(1(2)2')2)2}
{n,n(1(1(2)2\22)2)2} = {n,n(1(1(2)2/22)2)2}
{n,n,{n,3,2}}&n
170FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωωωωƐ0
φ(ωωωω,0) = φ(φ(φ(φ(φ(1)))),0)
ψ(Ωωωωω) = ψ011000(1))))))
ψ011000(1))))))
{n,n(1(1(1,2)2')2)2}
{n,n(1(1(1,2)2\22)2)2} = {n,n(1(1(1,2)2/22)2)2}
{n,n,{n,4,2}}&n
171FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωωωωωƐ0
φ(ωωωωω,0) = φ(φ(φ(φ(φ(φ(1))))),0)
ψ(Ωωωωωω) = ψ0110000(1)))))))
ψ0110000(1)))))))
{n,n(1(1(1(2)2)2')2)2}
{n,n(1(1(1(2)2)2\22)2)2} = {n,n(1(1(1(2)2)2/22)2)2}
{n,n,{n,5,2}}&n
172FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
1Ɛ0Ɛ0
φ(ε0,0) = φ(φ(1,0),0) = φ(φ(φ(0),0),0)
ψ(Ωψ(Ω)) = ψ01101(1))))) = ψ0110000(...)))))))
ψ01101(0)))))
{n,n(1(1.2')2)2} = {n,n(1(1(1'2)2')2)2}
{n,n(1(1\2\22)2)2} = {n,n(1(1(1/2)2/22)2)2}
{n,n,{n,n,2}}&n
173FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
2Ɛ0Ɛ0
φ(ζ0,0) = φ(φ(2,0),0)
ψ(Ωψ(Ω2))
ψ011011(0))))))
{n,n(1(1(1'1'2)2')2)2}
{n,n(1(1\1\2\22)2)2} = {n,n(1(1(1/1/2)2/22)2)2}
{n,n,{n,n,3}}&n
174FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
3Ɛ0Ɛ0
φ(η0,0) = φ(φ(3,0),0)
ψ(Ωψ(Ω3))
ψ011011(0)+ψ1(0))))))
{n,n(1(1(1'1'1'2)2')2)2}
{n,n(1(1\1\1\2\22)2)2} = {n,n(1(1(1/1/1/2)2/22)2)2}
{n,n,{n,n,4}}&n
175FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
4Ɛ0Ɛ0
φ(φ(4,0),0)
ψ(Ωψ(Ω4))
ψ011011(0)+ψ1(0)+ψ1(0))))))
{n,n(1(1(1'1'1'1'2)2')2)2}
{n,n(1(1\1\1\1\2\22)2)2} = {n,n(1(1(1/1/1/1/2)2/22)2)2}
{n,n,{n,n,5}}&n
176FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ωƐ0Ɛ0
φ(φ(ω,0),0)
ψ(Ωψ(Ωω))
ψ011011(1))))))
{n,n(1(1(1(2')2)2')2)2}
{n,n(1(1(2\22)2\22)2)2} = {n,n(1(1(1(2/22)2)2/22)2)2}
{n,n,{n,n,n}}&n
177FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
1Ɛ0Ɛ0Ɛ0
φ(φ(ε0,0),0) = φ(φ(φ(φ(0),0),0),0)
ψ(Ωψ(Ωψ(Ω)))
ψ01101101(0))))))))
{n,n(1(1(1(1.2')2)2')2)2} = {n,n(1(1(1(1(1'2)2')2)2')2)2}
{n,n(1(1(1\2\22)2\22)2)2} = {n,n(1(1(1(1(1/2)2/22)2)2/22)2)2}
{n,n,{n,n,{n,n,2}}}&n
178FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
1Ɛ0Ɛ0Ɛ0Ɛ0
φ(φ(φ(φ(φ(0),0),0),0),0)
ψ(Ωψ(Ωψ(Ωψ(Ω))))
ψ01101101101(0)))))))))))
{n,n(1(1(1(1(1(1(1'2)2')2)2')2)2')2)2}
{n,n(1(1(1(1\2\22)2\22)2\22)2)2} = {n,n(1(1(1(1(1(1(1/2)2/22)2)2/22)2)2/22)2)2}
{n,n,{n,n,{n,n,{n,n,2}}}}&n
179FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
1Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0
φ(φ(φ(φ(φ(φ(0),0),0),0),0),0)
ψ(Ωψ(Ωψ(Ωψ(Ωψ(Ω)))))
ψ01101101101101(0))))))))))))))
{n,n(1(1(1(1(1(1(1(1(1'2)2')2)2')2)2')2)2')2)2}
{n,n(1(1(1(1(1\2\22)2\22)2\22)2\22)2)2} = {n,n(1(1(1(1(1(1(1(1(1/2)2/22)2)2/22)2)2/22)2)2/22)2)2}
{n,n,{n,n,{n,n,{n,n,{n,n,2}}}}}&n
180name
proof theory
FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Feferman–Schütte ordinal
ATR0
Г0 = ...Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0
φ(1,0,0) = φ(φ(φ(φ(φ(φ(...,0),0),0),0),0),0) = φ(12)
ψ(ΩΩ) = ψ(Ωψ(Ωψ(Ωψ(Ωψ(...)))))
ψ0111(0)))) = ψ011011011(...)))))))))
{n,n(1(1'2')2)2} = {n,n(1(1(1...'...2)2')2)2} ~n insert
{n,n(1(1\23)2)2} = {n,n(1(1...\...2\22)2)2} ~n insert = {n,n(1(1/2/22)2)2} = {n,n(1(1.../...2/22)2)2} ~n insert
{n,n,1,2}&n
181FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0+1
φ(1,0,0)+1
ψ(ΩΩ)+1
ψ0111(0))))+ψ0(0)
{n,n(2(1'2')2)2}
{n,n(2(1\23)2)2} = {n,n(2(1/2/22)2)2}
182FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωГ0+1
φ(Г0+1) = φ(φ(1,0,0)+1)
ψ(ΩΩ+1)
ψ0111(0)))+ψ0(0))
{n,n(1,2(1'2')2)2}
{n,n(1,2(1\23)2)2} = {n,n(1,2(1/2/22)2)2}
183FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωωГ0+1
φ(φ(Г0+1)) = φ(φ(φ(1,0,0)+1))
ψ(ΩΩ+ψ(ΩΩ+1))
ψ0111(0)))+ψ0111(0)))+ψ0(0)))
{n,n(1(2)2(1'2')2)2}
{n,n(1(2)2(1\23)2)2} = {n,n(1(2)2(1/2/22)2)2}
184FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωωωГ0+1
φ(φ(φ(Г0+1))) = φ(φ(φ(φ(1,0,0)+1)))
ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+1)))
ψ0111(0)))+ψ0111(0)))+ψ0111(0)))+ψ0(0))))
{n,n(1(1,2)2(1'2')2)2}
{n,n(1(1,2)2(1\23)2)2} = {n,n(1(1,2)2(1/2/22)2)2}
185FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
εГ0+1 = ωωω...Г0+1
φ(1,Г0+1) = φ(φ(φ(φ(...φ(1,0,0)+1...))))
ψ(ΩΩ+Ω) = ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+...))))
ψ0111(0)))+ψ1(0))
{n,n(1'2(1'2')2)2}
{n,n(1\2(1\23)2)2} = {n,n(1/2(1/2/22)2)2}
186FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ζГ0+1 = εεε...Г0+1
φ(2,Г0+1) = φ(1,φ(1,φ(1,φ(1,...φ(1,0,0)+1...))))
ψ(ΩΩ2)
ψ0111(0)))+ψ11(0)))
{n,n(1'1'2(1'2')2)2}
{n,n(1\1\2(1\23)2)2} = {n,n(1/1/2(1/2/22)2)2}
187FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ηГ0+1 = ζζζ...Г0+1
φ(3,Г0+1) = φ(2,φ(2,φ(2,φ(2,...φ(1,0,0)+1...))))
ψ(ΩΩ3)
ψ0111(0)))+ψ11(0)+ψ1(0)))
{n,n(1'1'1'2(1'2')2)2}
{n,n(1\1\1\2(1\23)2)2} = {n,n(1/1/1/2(1/2/22)2)2}
188FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωƐГ0+1
φ(ω,Г0+1) = φ(φ(1,0),φ(1,0,0)+1)
ψ(ΩΩω)
ψ0111(0)))+ψ11(1)))
{n,n(1(2')2(1'2')2)2}
{n,n(1(2\22)2(1\23)2)2} = {n,n(1(2/22)2(1/2/22)2)2}
189FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωƐ0ƐГ0+1
φ(φ(ω,0),Г0+1) = φ(φ(φ(1,0),0),φ(1,0,0)+1)
ψ(ΩΩψ(Ωω))
ψ0111(0)))+ψ11011(1))))))
{n,n(1(1(1(2')2)2')2(1'2')2)2}
{n,n(1(1(2\22)2\22)2(1\23)2)2} = {n,n(1(1(2/22)2/22)2(1/2/22)2)2}
190FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωƐ0Ɛ0ƐГ0+1
φ(φ(φ(ω,0),0),Г0+1) =  φ(φ(φ(φ(1,0),0),0),φ(1,0,0)+1)
ψ(ΩΩψ(Ωψ(Ωω)))
ψ0111(0)))+ψ11011011(1)))))))))
{n,n(1(1(1(1(1(2')2)2')2)2')2(1'2')2)2}
{n,n(1(1(1(2\22)2\22)2\22)2(1\23)2)2} = {n,n(1(1(1(2/22)2/22)2/22)2(1/2/22)2)2}
191FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0Ɛ1
φ(Г0,1) = φ(Г00+1) = φ(φ(1,0,0),1) = φ(φ(1,0,0),φ(1,0,0)+1) = φ(φ(φ(...,0),0),φ(1,0,0)+1)
ψ(ΩΩψ(ΩΩ)) = ψ(ΩΩψ(Ωψ(Ωψ(Ω...))))
ψ0111(0)))+ψ110111(0)))))))
{n,n(1(1(1(1'2')2)2')2(1'2')2)2}
{n,n(1(1(1\23)2\22)2(1\23)2)2} = {n,n(1(1(1/2/22)2/22)2(1/2/22)2)2}
192FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0Ɛ2
φ(Г0,2) = φ(φ(1,0,0),2) = φ(φ(φ(...,0),0),φ(Г0,1)+1) = φ(φ(φ(...,0),0),φ(φ(1,0,0),1)+1)
ψ(ΩΩψ(ΩΩ)×2)
ψ0111(0)))+ψ110111(0)))))+ψ0(0)))
{n,n(1(1(1(1'2')2)2')3(1'2')2)2}
{n,n(1(1(1\23)2\22)3(1\23)2)2} = {n,n(1(1(1/2/22)2/22)3(1/2/22)2)2}
193FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0ƐГ0
φ(Г00) = φ(φ(1,0,0),φ(1,0,0))
ψ(ΩΩψ(ΩΩ)×ψ(ΩΩ))
ψ0111(0)))+ψ110111(0)))))+ψ0111(0))))))
{n,n(1(1(1(1'2')2)2')1(1(1'2')2)2(1'2')2)2}
{n,n(1(1(1\23)2\22)1(1\23)2(1\23)2)2} = {n,n(1(1(1/2/22)2/22)1(1/2/22)2(1/2/22)2)2}
194FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0ƐГ0ƐГ0
φ(Г0,φ(Г00)) = φ(φ(1,0,0),φ(φ(1,0,0),φ(1,0,0)))
ψ(ΩΩψ(ΩΩ)×ψ(ΩΩψ(ΩΩ)×ψ(ΩΩ))
ψ0111(0)))+ψ110111(0)))))+ψ0111(0)))+ψ110111(0)))))))))
{n,n(1(1(1(1'2')2)2')1(1(1(1(1'2')2)2')1(1(1'2')2)2(1'2')2)2(1'2')2)2}
{n,n(1(1(1\23)2\22)1(1(1\23)2\22)1(1\23)2(1\23)2)2(1\23)2)2} = {n,n(1(1(1/2/22)2/22)1(1(1/2/22)2/22)1(1/2/22)2(1/2/22)2)2(1/2/22)2)2}
195FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0+1Ɛ0
φ(Г0+1,0) = φ(φ(1,0,0)+1,0) = φ(Г0,φ(Г0,φ(Г0,φ(Г0,...)))) = φ(φ(1,0,0),φ(φ(1,0,0),φ(φ(1,0,0),φ(φ(1,0,0)...))))
ψ(ΩΩψ(ΩΩ)×Ω) = ψ(ΩΩψ(ΩΩ)+1) = ψ(ΩΩψ(ΩΩ)×ψ(ΩΩψ(ΩΩ)×ψ(ΩΩψ(ΩΩ)×...)))
ψ0111(0)))+ψ110111(0))))+ψ0(0))))
{n,n(1(1(1(1'2')2)2')1'2(1'2')2)2}
{n,n(1(1(1\23)2\22)1\2(1\23)2)2} = {n,n(1(1(1/2/22)2/22)1/2(1/2/22)2)2}
196FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0+1Ɛ0Ɛ0
φ(φ(Г0+1,0),0) = φ(φ(φ(1,0,0)+1,0),0)
ψ(ΩΩψ(ΩΩψ(ΩΩ)+1))
ψ0111(0)))+ψ110111(0)))+ψ0111(0)))+ψ110111(0))))+ψ0(0))))))))
{n,n(1(1(1(1(1(1'2')2)2')1'2(1'2')2)2')2(1'2')2)2}
{n,n(1(1(1(1\23)2\22)2(1\23)2\22)1\2(1\23)2)2} = {n,n(1(1(1(1/2/22)2/22)2(1/2/22)2/22)1/2(1/2/22)2)2}
197FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0+1Ɛ0Ɛ0Ɛ0
φ(φ(φ(Г0+1,0),0),0) = φ(φ(φ(φ(1,0,0)+1,0),0),0)
ψ(ΩΩψ(ΩΩψ(ΩΩψ(ΩΩ)+1)))
ψ0111(0)))+ψ110111(0)))+ψ0111(0)))+ψ110111(0)))+ψ0111(0)))+ψ110111(0))))+ψ0(0))))))))))))
{n,n(1(1(1(1(1(1(1(1'2')2)2')1'2(1'2')2)2')2(1'2')2)2')2(1'2')2)2}
{n,n(1(1(1(1(1\23)2\22)2(1\23)2\22)2(1\23)2\22)1\2(1\23)2)2} = {n,n(1(1(1(1(1/2/22)2/22)2(1/2/22)2/22)2(1/2/22)2/22)1/2(1/2/22)2)2}
198FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г1Г0+1...Ɛ0Ɛ0Ɛ0
φ(1,0,1) = φ(φ(φ(...φ(φ(1,0,0)+1, 0), ...,0),0),0) = φ(1210)
ψ(ΩΩ×2) = ψ(ΩΩΩ) = ψ(ΩΩψ(ΩΩψ(ΩΩψ(ΩΩ)+...)))
ψ0111(0)))+ψ111(0))))
{n,n(1(1'2')3)2}
{n,n(1(1\23)3)2} = {n,n(1(1/2/22)3)2}
199FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г2
φ(1,0,2) = φ(φ(φ(...φ(φ(1,0,1)+1, 0), ...,0),0),0) = φ(1220)
ψ(ΩΩ×3)
ψ0111(0)))+ψ111(0)))+ψ111(0))))
{n,n(1(1'2')4)2}
{n,n(1(1\23)4)2} = {n,n(1(1/2/22)4)2}
200FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Гω
φ(1,0,φ(1))
ψ(ΩΩ×ω)
ψ0111(0))+ψ0(0)))
{n,n(1(1'2')1,2)2}
{n,n(1(1\23)1,2)2} = {n,n(1(1/2/22)1,2)2}
201FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Гε0
φ(1,0,φ(1,0))
ψ(ΩΩ×ψ(Ω))
ψ0111(0))+ψ01(0))))
{n,n(1(1'2')1.2)2}
{n,n(1(1\23)1(1\2)2)2} = {n,n(1(1/2/22)1(1/2)2)2}
202FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ГГ0
φ(1,0,φ(1,0,0))
ψ(ΩΩ×ψ(ΩΩ))
ψ0111(0))+ψ0111(0))))))
{n,n(1(1'2')1(1(1'2')2)2)2}
{n,n(1(1\23)1(1(1\23)2)2)2} = {n,n(1(1/2/22)1(1(1/2/22)2)2)2}
203FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ГГГ0
φ(1,0,φ(1,0,φ(1,0,0)))
ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ))
ψ0111(0))+ψ0111(0))+ψ0111(0))))))))
{n,n(1(1'2')1(1(1'2')1(1(1'2')2)2)2)2}
{n,n(1(1\23)1(1(1\23)1(1(1\23)2)2)2)2} = {n,n(1(1/2/22)1(1(1/2/22)1(1(1/2/22)2)2)2)2}
204FGH
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ГГГГ...  = 1Г
φ(1,1,0) = φ(1,0,φ(1,0,φ(1,0,φ(1,0,...)))) = φ(2111)
ψ(ΩΩ×Ω) = ψ(ΩΩ+1)= ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ×...))
ψ0111(0))+ψ1(0)))
{n,n(1(1'2')1'2)2}
{n,n(1(1\23)1\2)2} = {n,n(1(1/2/22)1/2)2}
205FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
2Г
φ(1,2,0) = φ(1,1,φ(1,1,φ(1,1,φ(1,1,...)))) = φ(1221)
ψ(ΩΩ+2)
ψ0111(0))+ψ1(0)+ψ1(0)))
{n,n(1(1'2')1'1'2)2}
{n,n(1(1\23)1\1\2)2} = {n,n(1(1/2/22)1/1/2)2}
206FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ωГ
φ(1,ω,0) = φ(1,φ(1),0)
ψ(ΩΩ+ω)
ψ0111(0))+ψ1(1)))
{n,n(1(1'2')1(2')2)2}
{n,n(1(1\23)1(2\22)2)2} = {n,n(1(1/2/22)1(2/22)2)2}
207FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
1Ɛ0Г
φ(1,ε0,0) = φ(1,φ(1,0),0)
ψ(ΩΩ+ψ(Ω))
ψ0111(0))+ψ101(0)))))
{n,n(1(1'2')1(1.2')2)2}
{n,n(1(1\23)1(1\2\22)2)2} = {n,n(1(1/2/22)1(1/2/22)2)2}
208FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0Г0 = ...Ɛ0Ɛ0Ɛ0Ɛ0Г0
φ(1,Г0,0) = φ(1,φ(1,0,0),0)
ψ(ΩΩ+ψ(ΩΩ))
ψ0111(0))+ψ10111(0))))))
{n,n(1(1'2')1(1(1(1'2')2)2')2)2}
{n,n(1(1\23)1(1(1\23)2\22)2)2} = {n,n(1(1/2/22)1(1(1/2/22)2/22)2)2}
209FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0Г0Г0
φ(1,φ(1,φ(1,0,0),0),0)
ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ)))
ψ0111(0))+ψ10111(0))+ψ10111(0))))))))))
{n,n(1(1'2')1(1(1(1'2')1(1(1(1'2')2)2')2)2')2)2}
{n,n(1(1\23)1(1(1\23)1(1(1\23)2\22)2\22)2)2} = {n,n(1(1/2/22)1(1(1/2/22)1(1(1/2/22)2/22)2/22)2)2}
210FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
Г0Г0Г0Г0
φ(1,φ(1,φ(1,φ(1,0,0),0),0),0)
ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ))))
ψ0111(0))+ψ10111(0))+ψ10111(0))+ψ10111(0)))))))))))))
{n,n(1(1'2')1(1(1(1'2')1(1(1(1'2')1(1(1(1'2')2)2')2)2')2)2')2)2}
{n,n(1(1\23)1(1(1\23)1(1(1\23)1(1(1\23)2\22)2\22)2\22)2)2} = {n,n(1(1/2/22)1(1(1/2/22)1(1(1/2/22)1(1(1/2/22)2/22)2/22)2/22)2)2}
211FGH
FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
...Г0Г0Г0Г0
φ(2,0,0) = φ(1,φ(1,φ(1,φ(1,φ(1,...,0),0),0),0),0) = φ(22)
ψ(ΩΩ×2) = ψ(ΩΩ+Ω) = ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+...)))
ψ0111(0))+ψ11(0))))
{n,n(1(1'2')1(1'2')2)2}
{n,n(1(1\23)1(1\23)2)2} = {n,n(1(1/2/22)1(1/2/22)2)2}
{n,n,1,3}&n
212FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
φ(2,0,1) = φ(1,φ(1,φ(1,φ(1,...φ(2,0,0)+1...,0),0),0),0) = φ(2210)
ψ(ΩΩ×2×2)
ψ0111(0))+ψ11(0)))+ψ111(0))+ψ11(0))))
{n,n(1(1'2')1(1'2')3)2}
{n,n(1(1\23)1(1\23)3)2} = {n,n(1(1/2/22)1(1/2/22)3)2}
213FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
φ(2,1,0) = φ(2,0,φ(2,0,φ(2,0,φ(2,0,...)))) = φ(2211)
ψ(ΩΩ×2+1) = ψ(ΩΩ×2×Ω)
ψ0111(0))+ψ11(0))+ψ0(0)))
{n,n(1(1'2')1(1'2')1'2)2}
{n,n(1(1\23)1(1\23)1\2)2} = {n,n(1(1/2/22)1(1/2/22)1/2)2}
214FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(3,0,0) = φ(2,φ(2,φ(2,φ(2,...,0),0),0),0) = φ(32)
ψ(ΩΩ×3)
ψ0111(0))+ψ11(0))+ψ11(0))))
{n,n(1(1'2')1(1'2')1(1'2')2)2}
{n,n(1(1\23)1(1\23)1(1\23)2)2} = {n,n(1(1/2/22)1(1/2/22)1(1/2/22)2)2}
{n,n,1,4}&n
215FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(4,0,0) = φ(42)
ψ(ΩΩ×4)
ψ0111(0))+ψ11(0))+ψ11(0))+ψ11(0))))
{n,n(1(1'2')1(1'2')1(1'2')1(1'2')2)2}
{n,n(1(1\23)1(1\23)1(1\23)1(1\23)2)2} = {n,n(1(1/2/22)1(1/2/22)1(1/2/22)1(1/2/22)2)2}
{n,n,1,5}&n
216FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(ω,0,0) = φ(φ(1),0,0) = φ(ω2)
ψ(ΩΩ×ω)
ψ0111(0)+ψ0(0))))
{n,n(1(2'2')2)2}
{n,n(1(2\23)2)2} = {n,n(1(2/2/22)2)2}
{n,n,1,n}&n
217FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(ω2,0,0) = φ(φ(2),0,0)
ψ(ΩΩ×ω2)
ψ0111(0)+ψ0(1))))
{n,n(1(3'2')2)2}
{n,n(1(3\23)2)2} = {n,n(1(3/2/22)2)2}
{n,n,1,{n,2}}&n
218FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(ωω,0,0) = φ(φ(φ(1)),0,0)
ψ(ΩΩ×ωω)
ψ0111(0)+ψ00(1)))))
{n,n(1(1,2'2')2)2}
{n,n(1(1,2\23)2)2} = {n,n(1(1,2/2/22)2)2}
{n,n,1,{n,2,2}}&n
219FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(ε0,0,0) = φ(φ(1,0),0,0)
ψ(ΩΩ×ψ(Ω))
ψ0111(0)+ψ01(0)))))
{n,n(1(1.2'2')2)2}
{n,n(1(1\2\23)2)2} = {n,n(1(1(1/2)2/2/22)2)2}
{n,n,1,{n,n,2}}&n
220FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(Г0,0,0) = φ(φ(1,0,0),0,0)
ψ(ΩΩ×ψ(ΩΩ))
ψ0111(0)+ψ0111(0)))))))
{n,n(1(1(1(1'2')2)2'2')2)2}
{n,n(1(1(1\23)2\23)2)2} = {n,n(1(1(1(1/2/22)2)2/2/22)2)2}
{n,n,1,{n,n,1,2}}&n
221FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(φ(φ(1,0,0),0,0),0,0)
ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ)))
ψ0111(0)+ψ0111(0)))))))
{n,n(1(1(1(1(1(1'2')2)2'2')2)2'2')2)2}
{n,n(1(1(1(1\23)2\23)2\23)2)2} = {n,n(1(1(1(1(1(1/2/22)2)2/2/22)2)2/2/22)2)2}
{n,n,1,{n,n,1,{n,n,1,2}}}&n
222FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(φ(φ(φ(1,0,0),0,0),0,0),0,0)
ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ))))
ψ0111(0)+ψ0111(0)+ψ0111(0))))))))))
{n,n(1(1(1(1(1(1(1(1'2')2)2'2')2)2'2')2)2'2')2)2}
{n,n(1(1(1(1(1\23)2\23)2\23)2\23)2)2} = {n,n(1(1(1(1(1(1(1(1/2/22)2)2/2/22)2)2/2/22)2)2/2/22)2)2}
{n,n,1,{n,n,1,{n,n,1,{n,n,1,2}}}}&n
223name
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Ackermann ordinal
φ(1,0,0,0) = φ(φ(φ(φ(...),0,0),0,0),0,0) = φ(13)
ψ(ΩΩ2) = ψ(ΩΩ×Ω) = ψ(ΩΩ×ψ(ΩΩ×ψ(ΩΩ×...)))
ψ0111(0)+ψ1(0))))
{n,n(1(1'3')2)2}
{n,n(1(1\24)2)2} = {n,n(1(1/3/22)2)2}
{n,n,1,1,2}&n
224FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1,0,0,0,0) = φ(14)
ψ(ΩΩ3)
ψ0111(0)+ψ1(0)+ψ1(0))))
{n,n(1(1'4')2)2}
{n,n(1(1\25)2)2} = {n,n(1(1/4/22)2)2}
{n,n,1,1,1,2}&n
225name
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Small Veblen ordinal
φ(1ω) = φ(1φ(10)) = φ(1,0,0,0,0,...)
ψ(ΩΩω)
ψ0111(1)))) = ψ0111(0)+ψ1(0)+ψ1(0)+ψ1(0)+...)))
{n,n(1(1'1,2')2)2}
{n,n(1(1\21,2)2)2} = {n,n(1(1/1,2/22)2)2}
{n,n(2)2}&n = n&n&n
226FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1ω+1)
ψ(ΩΩω+1)
ψ0111(1)+ψ1(0))))
{n,n(1(1'2,2')2)2}
{n,n(1(1\22,2)2)2} = {n,n(1(1/2,2/22)2)2}
{n,n(2)n}&n
227FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1ω×2)
ψ(ΩΩω×2)
ψ0111(1)+ψ1(1))))
{n,n(1(1'1,3')2)2}
{n,n(1(1\21,3)2)2} = {n,n(1(1/1,3/22)2)2}
{n,n(2)n(2)n}&n
228FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1ω2)
ψ(ΩΩω2)
ψ0111(2))))
{n,n(1(1'1,1,2')2)2}
{n,n(1(1\21,1,2)2)2} = {n,n(1(1/1,1,2/22)2)2}
{n,n(3)n}&n
229FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1ωω)
ψ(ΩΩωω)
ψ01110(1)))))
{n,n(1(1'1(2)2')2)2}
{n,n(1(1\21(2)2)2)2} = {n,n(1(1/1(2)2/22)2)2}
{n,n(1,2)n}&n
230FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1ε0) = φ(1φ(11))
ψ(ΩΩψ(Ω))
ψ011101(0))))))
{n,n(1(1'1.2')2)2}
{n,n(1(1\21\2)2)2} = {n,n(1(1/1(1/2)2/22)2)2}
{n,n,2}&n&n
231FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1Г0) = φ(1φ(12))
ψ(ΩΩψ(ΩΩ))
ψ01110111(0))))))))
{n,n(1(1'1(1(1'2')2)2')2)2}
{n,n(1(1\21(1\23)2)2)2} = {n,n(1(1/1(1(1/2/22)2)2/22)2)2}
{n,n,1,2}&n&n
232FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1φ(1ω)) = φ(1φ(1φ(10)))
ψ(ΩΩψ(ΩΩω))
ψ01110111(1))))))))
{n,n(1(1'1(1(1'1,2')2)2')2)2}
{n,n(1(1\21(1\21,2)2)2)2} = {n,n(1(1/1(1(1/1,2/22)2)2/22)2)2}
{n,n(2)2}&n&n = n&n&n&n
233FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1φ(1φ(12)))
ψ(ΩΩψ(ΩΩψ(ΩΩ)))
ψ011101110111(0))))))))))))
{n,n(1(1'1(1(1'1(1(1'1'2')2)2')2)2')2)2}
{n,n(1(1\21(1\21(1\23)2)2)2)2} = {n,n(1(1/1(1(1/1(1(1/2/22)2)2/22)2)2/22)2)2}
{n,n,1,2}&n&n&n
234FGH Veblen φ
FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
φ(1φ(1φ(1ω))) = φ(1φ(1φ(1φ(10))))
ψ(ΩΩψ(ΩΩψ(ΩΩω)))
ψ011101110111(1))))))))))))
{n,n(1(1'1(1(1'1(1(1'1,2')2)2')2)2')2)2}
{n,n(1(1\21(1\21(1\21,2)2)2)2)2} = {n,n(1(1/1(1(1/1(1(1/1,2/22)2)2/22)2)2/22)2)2}
n&n&n&n&n
235name
FGH Veblen φ

FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
Large Veblen ordinal
φ(1φ(1φ(1φ(1...))))
ψ(ΩΩΩ) = ψ(ΩΩψ(ΩΩψ(ΩΩ...)))
ψ01111(0)))))
{n,n(1(1'1'2')2)2}
{n,n(1(1\21\22)2)2} = {n,n(1(1/1/2/22)2)2}
{n,n/2} = {n,n{L,1}2} = n&n ~n+1 repeat
236FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)×2
ψ01111(0)))))+ψ01111(0)))))
{n,n(1(1'1'2')2)3}
{n,n(1(1\21\22)2)3} = {n,n(1(1/1/2/22)2)3}
{n,n/3}
237FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)×3
ψ01111(0)))))+ψ01111(0)))))+ψ01111(0)))))
{n,n(1(1'1'2')2)4}
{n,n(1(1\21\22)2)4} = {n,n(1(1/1/2/22)2)4}
{n,n/4}
238FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)×ω = ψ(ΩΩΩ+1)
ψ01111(0))))+ψ0(0))
{n,n(1(1'1'2')2)1,2}
{n,n(1(1\21\22)2)1,2} = {n,n(1(1/1/2/22)2)1,2}
{n,n/n}
239FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)2
ψ01111(0))))+ψ01111(0))))))
{n,n(1(1'1'2')2)1(1'1'2')2}
{n,n(1(1\21\22)2)1(1\21\22)2} = {n,n(1(1/1/2/22)2)1(1/1/2/22)2}
{n,n/{n,n/2}}= {n,n/1/2}
240FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)3
ψ01111(0))))+ψ01111(0)))))+ψ01111(0))))))
{n,n(1(1'1'2')2)1(1'1'2')1(1'1'2')2}
{n,n(1(1\21\22)2)1(1\21\22)1(1\21\22)2} = {n,n(1(1/1/2/22)2)1(1/1/2/22)1(1/1/2/22)2}
{n,n/1/1/2}
241FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)ω = ψ(ΩΩΩ+ψ(ΩΩΩ+1))
ψ01111(0))))+ψ01111(0))))+ψ0(0)))
{n,n(2(1'1'2')2)2}
{n,n(2(1\21\22)2)2} = {n,n(2(1/1/2/22)2)2}
{n,n(/2)2} = n&&n = {n,n/12} ~n+1 repeat
242FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)ψ(ΩΩΩ) = ψ(ΩΩΩ+ψ(ΩΩΩ+ψ(ΩΩΩ))
ψ01111(0))))+ψ01111(0))))+ψ01111(0)))))))
{n,n(1(2)2(1'1'2')2)2}
{n,n(1(2)2(1\21\22)2)2} = {n,n(1(2)2(1/1/2/22)2)2}
{n,n(/2)2}&&n = n&&n&&n
243FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)ψ(ΩΩΩ)ψ(ΩΩΩ= ψ(ΩΩΩ+ψ(ΩΩΩ+ψ(ΩΩΩ)+ψ(ΩΩΩ))
ψ01111(0))))+ψ01111(0))))+ψ01111(0))))+ψ01111(0))))))))
{n,n(1(1(2)2)2(1'1'2')2)2}
{n,n(1(1(2)2)2(1\21\22)2)2} = {n,n(1(1(2)2)2(1/1/2/22)2)2}
n&&n&&n&&n
244FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω)
ψ01111(0))))+ψ1(0))
{n,n(1'2(1'1'2')2)2}
{n,n(1\2(1\21\22)2)2} = {n,n(1/2(1/1/2/22)2)2}
{n,n//2} = {n,n{L,2}2} = n&&n ~n repeat
245FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω+1)
ψ01111(0))))+ψ1(0)+ψ0(0))
{n,n(1'3(1'1'2')2)2}
{n,n(1\3(1\21\22)2)2} = {n,n(1/3(1/1/2/22)2)2}
{n,n///2} = {n,n{L,3}2}
246FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω+ω)
ψ01111(0))))+ψ1(0)+ψ0(1))
{n,n(1'1,2(1'1'2')2)2}
{n,n(1\1,2(1\21\22)2)2} = {n,n(1/1,2(1/1/2/22)2)2}
{n,n(2)/2} = {n,n{L,n}2}
247FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω+ω2)
ψ01111(0))))+ψ1(0)+ψ0(2))
{n,n(1'1,1,2(1'1'2')2)2}
{n,n(1\1,1,2(1\21\22)2)2} = {n,n(1/1,1,2(1/1/2/22)2)2}
{n,n{L,{n,2}}2}
248FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω+ωω)
ψ01111(0))))+ψ1(0)+ψ00(1)))
{n,n(1'1(2)2(1'1'2')2)2}
{n,n(1\1(2)2(1\21\22)2)2} = {n,n(1/1(2)2(1/1/2/22)2)2}
{n,n(1,2)/2} = {n,n{L,{n,n}}2}
249FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω+ψ(Ω))
ψ01111(0))))+ψ1(0)+ψ01(0)))
{n,n(1'1(1'2)2(1'1'2')2)2}
{n,n(1\1(1\2)2(1\21\22)2)2} = {n,n(1/1(1/2)2(1/1/2/22)2)2}
{n,n{L,{n,n,2}2}
250FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω×2)
ψ01111(0))))+ψ1(0)+ψ1(0))
{n,n(1'1(1(1'1'2')2)2(1'1'2')2)2}
{n,n(1\1(1(1\21\22)2)2(1\21\22)2)2} = {n,n(1/1(1(1/1/2/22)2)2(1/1/2/22)2)2}
{n,n{L,{n,n/2}2}
251FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+Ω×ψ(Ω))
ψ01111(0))))+ψ101(0))))
{n,n(1'1(2(1'1'2')2)2(1'1'2')2)2}
{n,n(1\1(2(1\21\22)2)2(1\21\22)2)2} = {n,n(1/1(2(1/1/2/22)2)2(1/1/2/22)2)2}
{n,n{L,{n,n(2)/2}2} = {n,n{L,L}2}
252FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ2)
ψ01111(0))))+ψ11(0)))
{n,n(1'1'2(1'1'2')2)2}
{n,n(1\1\2(1\21\22)2)2} = {n,n(1/1/2(1/1/2/22)2)2}
{n,n{L,L,n}2}
253FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ3)
ψ01111(0))))+ψ11(0)+ψ1(0)))
{n,n(1'1'1'2(1'1'2')2)2}
{n,n(1\1\1\2(1\21\22)2)2} = {n,n(1/1/1/2(1/1/2/22)2)2}
{n,n{L,L,L,n}2}
254FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ4)
ψ01111(0))))+ψ11(0)+ψ1(0)+ψ1(0)))
{n,n(1'1'1'1'2(1'1'2')2)2}
{n,n(1\1\1\1\2(1\21\22)2)2} = {n,n(1/1/1/1/2(1/1/2/22)2)2}
{n,n{L,L,L,L,n}2}
255FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩω)
ψ01111(0))))+ψ11(1)))
{n,n(1(2')2(1'1'2')2)2}
{n,n(1(2\22)2(1\21\22)2)2} = {n,n(1(2/22)2(1/1/2/22)2)2}
{n,n{L,L(2)2}2}
256FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩ)
ψ01111(0))))+ψ111(0))))
{n,n(1(1'2')2(1'1'2')2)2}
{n,n(1(1\23)2(1\21\22)2)2} = {n,n(1(1/2/22)2(1/1/2/22)2)2}
{n,n{L,L(1,2)2}2}
257FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩ2)
ψ01111(0))))+ψ111(0)+ψ1(0))))
{n,n(1(1'3')2(1'1'2')2)2}
{n,n(1(1\24)2(1\21\22)2)2} = {n,n(1(1/3/22)2(1/1/2/22)2)2}
{n,n{L,L(1,1,2)2}2}
258FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩω)
ψ01111(0))))+ψ111(1))))
{n,n(1(1'1,2')2(1'1'2')2)2}
{n,n(1(1\21,2)2(1\21\22)2)2} = {n,n(1(1/1,2/22)2(1/1/2/22)2)2}
{n,n{L,L(1(2)2)2}2} = {n,n{L,L(2)L}2}
259FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩψ(ΩΩω))
ψ01111(0))))+ψ1110111(1))))))))
{n,n(1(1'1(1(2')2)2')2(1'1'2')2)2}
{n,n(1(1\21(1(2\22)2\22) 2)2(1\21\22)2)2} = {n,n(1(1/1(1(2/22)2/22) 2/22)2(1/1/2/22)2)2}
{n,n{L,L(1(1(2)2)2)2}2}
260FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩψ(ΩΩψ(ΩΩω)))
ψ01111(0))))+ψ11101110111(1))))))))))))
{n,n(1(1'1(1(1(1(2')2)2')2)2')2(1'1'2')2)2}
{n,n(1(1\21(1(1(2\22)2\22)2\22)2)2(1\21\22)2)2} = {n,n(1(1/1(1(1(2/22)2/22)2/22)2/22)2(1/1/2/22)2)2}
{n,n{L,L(1(1(1(2)2)2)2)2}2}
261FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩψ(ΩΩΩ))
ψ01111(0))))+ψ11101111(0)))))))))
{n,n(1(1'1(1(1'2')2)2')2(1'1'2')2)2}
{n,n(1(1\21(1(1\23)2\22) 2)2(1\21\22)2)2} = {n,n(1(1/1 (1(1/2/22)2/22)2/22)2(1/1/2/22)2)2}
{n,n{L,L(2){L,L(2)L}}2}
262FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩψ(ΩΩΩΩω))
ψ01111(0))))+ψ11101111(0))))+ψ111(1))))))))
{n,n(1(1'1(1(1'1,2')2)2')2(1'1'2')2)2}
{n,n(1(1\21(1\21,2)2)2(1\21\22)2)2} = {n,n(1(1/1(1(1/1,2/22)2)2/22)2(1/1/2/22)2)2}
{n,n{L,L(3)L}2}
263FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩψ(ΩΩΩΩψ(ΩΩΩΩω)))
ψ01111(0))))+ψ11101111(0))))+ψ11101111(0))))+ψ111(1))))))))))))
{n,n(1(1'1(1(1'1(1(1'1,2')2)2')2)2')2(1'1'2')2)2}
{n,n(1(1\21(1\21(1\21,2)2)2)2(1\21\22)2)2} = {n,n(1(1/1(1(1/1(1(1/1,2/22)2)2/22)2)2/22)2(1/1/2/22)2)2}
{n,n{L,L(4)L}2}
264FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩΩ) = ψ(ΩΩΩ×2)
ψ01111(0))))+ψ1111(0)))))
{n,n(1(1'1'2')3)2}
{n,n(1(1\21\22)3)2} = {n,n(1(1/1/2/22)3)2}
{n,n{L,L(n)L}2} = {n,n\2} = {n,n{L2,1}2}
265FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+Ω)
ψ01111(0))))+ψ1111(0))))+ψ1(0))
{n,n(1'2(1'1'2')3)2}
{n,n(1\2(1\21\22)3)2}= {n,n(1/2(1/1/2/22)2)2}
{n,n\\2} = {n,n{L2,2}2}
266FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+Ω+1)
ψ01111(0))))+ψ1111(0))))+ψ1(0)+ψ0(0))
{n,n(1'3(1'1'2')3)2}
{n,n(1\3 (1\21\22)3)2} = {n,n(1/3(1/1/2/22)3)2}
{n,n\\\3} = {n,n{L2,3}2}
267FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+Ω+ω)
ψ01111(0))))+ψ1111(0))))+ψ1(0)+ψ0(1))
{n,n(1'1,2(1'1'2')3)2}
{n,n(1\1,2 (1\21\22)3)2} = {n,n(1/1,2(1/1/2/22)3)2}
{n,n{L2,n}2}
268FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+Ω2)
ψ01111(0))))+ψ1111(0))))+ψ11(0)))
{n,n(1'1'2(1'1'2')3)2}
{n,n(1\1\2(1\21\22)3)2} = {n,n(1/1/2(1/1/2/22)3)2}
{n,n{L2,L2,n}2}
269FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+ΩΩ)
ψ01111(0))))+ψ1111(0))))+ψ111(0))))
{n,n(1(1'2')2(1'1'2')3)2}
{n,n(1(1\23)2(1\21\22)3)2} = {n,n(1(1/2/22)2(1/1/2/22)3)2}
{n,n{L2,L2(1,2)2}2}
270FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×2+ΩΩω)
ψ01111(0))))+ψ1111(0))))+ψ111(1))))
{n,n(1(1'1,2')2(1'1'2')3)2}
{n,n (1(1\21,2)2(1\21\22)3)2} = {n,n(1(1/1,2/22)2(1/1/2/22)3)2}
{n,n{L2,L2(2)L}2}
271FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×3)
ψ01111(0))))+ψ1111(0))))+ψ1111(0)))))
{n,n(1(1'1'2')4)2}
{n,n(1(1\21\22)4)2} = {n,n(1(1/1/2/22)4)2}
{n,n{L2,L2(n)L2}2} = {n,n{L3,1}2}
272FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×4)
ψ01111(0))))+ψ1111(0))))+ψ1111(0))))+ψ1111(0)))))
{n,n(1(1'1'2')5)2}
{n,n(1(1\21\22)5)2} = {n,n(1(1/1/2/22)5)2}
{n,n{L4,1}2}
273FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ω)
ψ01111(0)))+ψ0(0)))
{n,n(1(1'1'2')1,2)2}
{n,n(1(1\21\22)1,2)2} = {n,n(1(1/1/2/22)1,2)2}
{n,n{Ln}2}
274FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ω2)
ψ01111(0)))+ψ0(1)))
{n,n(1(1'1'2')1,1,2)2}
{n,n(1(1\21\22)1,1,2)2} = {n,n(1(1/1/2/22)1,1,2)2}
{n,n{L{n,2}2}
275FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ωω)
ψ01111(0)))+ψ00(0))))
{n,n(1(1'1'2')1(2)2)2}
{n,n(1(1\21\22)1(2)2)2} = {n,n(1(1/1/2/22)1(2)2)2}
{n,n{L{n,n}2}
276FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ψ(Ω))
ψ01111(0)))+ψ01(0))))
{n,n(1(1'1'2')1(1'2)2)2}
{n,n(1(1\21\22)1(1\2)2)2} = {n,n(1(1/1/2/22)1(1/2)2)2}
{n,n{L{n,n,2}2}
277FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ψ(ΩΩ))
ψ01111(0)))+ψ0111(0))))))
{n,n(1(1'1'2')1(1(1'2')2)2)2}
{n,n(1(1\21\22)1(1(1\23)2)2)2} = {n,n(1(1/1/2/22)1(1(1/2/22)2)2)2}
{n,n{L{n,n,1,2}2}
278FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ψ(ΩΩΩ))
ψ01111(0)))+ψ01111(0)))))))
{n,n(1(1'1'2')1(1(1'1'2')2)2)2}
{n,n(1(1\21\22)1(1(1\21\22)2)2)2} = {n,n(1(1/1/2/22)1(1(1/1/2/22)2)2)2}
{n,n{L{n,n/2}2} = {n,n{LL}2}
279FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ψ(ΩΩΩ×ψ(ΩΩΩ)))
ψ01111(0)))+ψ01111(0)))+ψ01111(0)))))))))
{n,n(1(1'1'2')1(1(1'1'2')1(1(1'1'2')2)2)2)2}
{n,n(1(1\21\22)1(1(1\21\22)1(1(1\21\22)2)2)2)2} = {n,n(1(1/1/2/22)1(1(1/1/2/22)1(1(1/1/2/22)2)2)2)2}
{n,n{LL{n,n/2}2} ={n,n{LLL}2}
280FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×Ω) = ψ(ΩΩΩ+1)
ψ01111(0)))+ψ1(0)))
{n,n(1(1'1'2')1'2)2}
{n,n(1(1\21\22)1\2)2} = {n,n(1(1/1/2/22)1/2)2}
{n,n{(n)L}2} = {n,n{L}2} ~n repeat
281FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)
ψ01111(0)))+ψ1(1)))
{n,n(1(1'1'2')1(2')2)2}
{n,n(1(1\21\22)1(2\22)2)2} = {n,n(1(1/1/2/22)1(2/22)2)2}
{n,n{({n,2})L}2}
282FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩω)
ψ01111(0)))+ψ10(1))))
{n,n(1(1'1'2')1(1,2')2)2}
{n,n(1(1\21\22)1(1,2\22)2)2} = {n,n(1(1/1/2/22)1(1,2/22)2)2}
{n,n{({n,n})L}2}
283FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+ψ(Ω))
ψ01111(0)))+ψ101(0)))))
{n,n(1(1'1'2')1(1.2')2)2}
{n,n(1(1\21\22)1(1\2\22)2)2} = {n,n(1(1/1/2/22)1(1(1/2)2/22)2)2}
{n,n{({n,n,2})L}2}
284FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+ψ(ΩΩ))
ψ01111(0)))+ψ10111(0)))))))
{n,n(1(1'1'2')1(1(1(2')2)2')2)2}
{n,n(1(1\21\22)1(1(2\22)2\22)2)2} = {n,n(1(1/1/2/22)1(1(1(2/22)2)2/22)2)2}
{n,n{({n,n,n})L}2}
285FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ+ψ(ΩΩΩ))
ψ01111(0)))+ψ101111(0))))))))
{n,n(1(1'1'2')1(1(1(1.2')2)2')2)2}
{n,n(1(1\21\22)1(1(1\2\22)2\22)2)2} = {n,n(1(1/1/2/22)1(1(1(1(1/2)2/22)2)2/22)2)2}
{n,n{({n,n,{n,n,n}})L}2}
286FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ)
ψ01111(0)))+ψ11(0))))
{n,n(1(1'1'2')1(1'2')2)2}
{n,n(1(1\21\22)1(1\23)2)2} = {n,n(1(1/1/2/22)1(1/2/22)2)2}
{n,n{({n,n,1,2})L}2}
287FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ2)
ψ01111(0)))+ψ111(0)))))
{n,n(1(1'1'2')1(1'3')2)2}
{n,n(1(1\21\22)1(1\24)2)2} = {n,n(1(1/1/2/22)1(1/3/22)2)2}
{n,n{({n,n,1,1,2})L}2}
288FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩω)
ψ01111(0)))+ψ111(1)))))
{n,n(1(1'1'2')1(1'1,2')2)2}
{n,n(1(1\21\22)1(1\21,2)2)2} = {n,n(1(1/1/2/22)1(1/1,2/22)2)2}
{n,n{({n,n(2)2})L}2}
289FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩΩ) = ψ(ΩΩΩ×2)
ψ01111(0)))+ψ1111(0))))))
{n,n(1(1'1'2')1(1'1'2')2)2}
{n,n(1(1\21\22)1(1\21\22)2)2} = {n,n(1(1/1/2/22)1(1/1/2/22)2)2}
{n,n{({n,n/2})L}2} = {n,n{(L)L}2}
290FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×3)
ψ01111(0)))+ψ1111(0)))))+ψ1111(0))))))
{n,n(1(1'1'2')1(1'1'2')1(1'1'2')2)2}
{n,n(1(1\21\22)1(1\21\22)1(1\21\22)2)2} = {n,n(1(1/1/2/22)1(1/1/2/22)1(1/1/2/22)2)2}
{n,n{((L)L)L}2}
291FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
BEAF
ψ(ΩΩΩ×ω)
ψ01111(0))+ψ0(0))))
{n,n(1(2'1'2')2)2}
{n,n(1(2\21\22)2} = {n,n(1(2/1/2/22)2)2}
{n,n{(...)L}2} ~n insert
292FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ω2)
ψ01111(0))+ψ0(1))))
{n,n(1(3'1'2')2)2}
{n,n(1(3\21\22)2)2} = {n,n(1(3/1/2/22)2)2}
293FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ψ(ΩΩΩ))
ψ01111(0))+ψ01111(0))))))))
{n,n(1(1(1(1'1'2')2)2'1'2')2)2}
{n,n(1(1(1\21\22)2\21\22)2)2} = {n,n(1(1(1(1/1/2/22)2)2/1/2/22)2)2}
294FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ψ(ΩΩΩ×ψ(ΩΩΩ)))
ψ01111(0))+ψ01111(0))+ψ01111(0)))))))))))
{n,n(1(1(1(1(1(1'1'2')2)2'1'2')2)2'1'2')2)2}
{n,n(1(1(1(1\21\22)2\21\22)2\21\22)2)2} = {n,n(1(1(1(1(1(1/1/2/22)2)2/1/2/22)2)2/1/2/22)2)2}
295FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×Ω) = ψ(ΩΩΩ+1)
ψ01111(0))+ψ1(0))))
{n,n(1(1'2'2')2)2}
{n,n(1(1\22\22)2)2} = {n,n(1(1/2/2/22)2)2}
296FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ+2)
ψ01111(0))+ψ1(0)+ψ1(0))))
{n,n(1(1'3'2')2)2}
{n,n(1(1\23\22)2)2} = {n,n(1(1/3/2/22)2)2}
297FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ+ψ(ΩΩΩ))
ψ01111(0))+ψ101111(0)))))))))
{n,n(1(1'1(1(1'1'2')2)2'2')2)2}
{n,n(1(1\21(1\21\22)2\22)2)2} = {n,n(1(1/1(1(1/1/2/22)2)2/2/22)2)2}
298FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ+ψ(ΩΩΩ+ψ(ΩΩΩ)))
ψ01111(0))+ψ101111(0))+ψ01111(0))))))))))))
{n,n(1(1'1(1(1'1(1(1'1'2')2)2'2')2)2'2')2)2}
{n,n(1(1\21(1\21(1\21\22)2\22)2\22)2)2} = {n,n(1(1/1(1(1/1(1(1/1/2/22)2)2/2/22)2)2/2/22)2)2}
299FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ+Ω) = ψ(ΩΩΩ×2)
ψ01111(0))+ψ11(0)))))
{n,n(1(1'1'3')2)2}
{n,n(1(1\21\23)2)2} = {n,n(1(1/1/3/22)2)2}
300FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×3)
ψ01111(0))+ψ11(0))+ψ11(0)))))
{n,n(1(1'1'1,2')2)2}
{n,n(1(1\21\21,2)2)2} = {n,n(1(1/1/1,2/22)2)2}
301FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ω)
ψ01111(0)+ψ0(0))))) = ψ01111(0))+ψ11(0))+ψ11(0))+ψ11(0))+...)))
{n,n(1(1'1'1,2')2)2}
{n,n(1(1\21\21,2)2)2} = {n,n(1(1/1/1,2/22)2)2}
302FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ψ(ΩΩΩ))
ψ01111(0)+ψ01111(0)))))))))
{n,n(1(1'1'1(1(1'1'2')2)2')2)2}
{n,n(1(1\21\21(1\21\22)2)2)2} = {n,n(1(1/1/1(1(1/1/2/22)2)2/22)2)2}
303FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ×ψ(ΩΩΩ×ψ(ΩΩΩ)))
ψ01111(0)+ψ01111(0)+ψ01111(0)))))))))))))
{n,n(1(1'1'1(1(1'1'1(1(1'1'2')2)2')2)2')2)2}
{n,n(1(1\21\21(1\21\21(1\21\22)2)2)2)2} = {n,n(1(1/1/1(1(1/1/1(1(1/1/2/22)2)2/22)2)2/22)2)2}
304FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ2)
ψ01111(0)+ψ1(0)))))
{n,n(1(1'1'1'2')2)2}
{n,n(1(1\21\21\22)2)2} = {n,n(1(1/1/1/2/22)2)2}
305FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ2+1)
ψ01111(0)+ψ1(0))+ψ0(0))))
{n,n(1(1'2'1'2')2)2}
{n,n(1(1\22\21\22)2)2} = {n,n(1(1/2/1/2/22)2)2}
306FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ2)
ψ01111(0)+ψ1(0))+ψ1(0))))
{n,n(1(1'1'2'2')2)2}
{n,n(1(1\21\22\22)2)2} = {n,n(1(1/1/2/2/22)2)2}
307FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ2×2)
ψ01111(0)+ψ1(0))+ψ11(0)+ψ1(0)))))
{n,n(1(1'1'1'3')2)2}
{n,n(1(1\21\21\23)2)2} = {n,n(1(1/1/1/3/22)2)2}
308FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ3)
ψ01111(0)+ψ1(0)+ψ1(0)))))
{n,n(1(1'1'1'1'2')2)2}
{n,n(1(1\21\21\21\22)2)2} = {n,n(1(1/1/1/1/2/22)2)2}
309FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ4)
ψ01111(0)+ψ1(0)+ψ1(0)+ψ1(0)))))
{n,n(1(1'1'1'1'1'2')2)2}
{n,n(1(1\21\21\21\21\22)2)2} = {n,n(1(1/1/1/1/1/2/22)2)2}
310FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩω)
ψ01111(1))))) = ψ01111(0)+ψ1(0)+ψ1(0)+ψ1(0)+...))))
{n,n(1(1(2')2')2)2}
{n,n(1(2\32)2)2} = {n,n(1(1(2/22)2/22)2)2}
311FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩψ(ΩΩΩ)) = ψ(ΩΩΩ↑ψ(ΩΩΩ))
ψ0111101111(0))))))))))
{n,n(1(1(1'1'2')2(2')2')2)2}
{n,n(1(1(1(1\21\22)2\32)2)2)2} = {n,n(1(1(1/1/2/22)2(2/22)2/22)2)2}
312FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩ↑ψ(ΩΩΩ↑ψ(ΩΩΩ)))
ψ011110111101111(0)))))))))))))))
{n,n(1(1(1(1'1'2')2(2')2')2(2')2')2)2}
{n,n(1(1(1(1(1(1\21\22)2\32)2)2\32)2)2)2} = {n,n(1(1(1(1/1/2/22)2(2/22)2/22)2(2/22)2/22)2)2}
313FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩΩ) = ψ(Ω↑↑4)
ψ011111(0))))))
{n,n(1(1(1'2')2')2)2}
{n,n(1(1\33)2)2} = {n,n(1(1(1/2/22)2/22)2)2}
314FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩΩΩ) = ψ(Ω↑↑5)
ψ0111111(0)))))))
{n,n(1(1(1'1'2')2')2)2}
{n,n(1(1(1\31\32)2)2)2} = {n,n(1(1(1/1/2/22)2/22)2)2}
315FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(ΩΩΩΩΩΩ) = ψ(Ω↑↑6)
ψ01111111(0))))))))
{n,n(1(1(1(1'2')2')2')2)2}
{n,n(1(1(1\43)2)2)2} = {n,n(1(1(1(1/2/22)2/22)2/22)2)2}
316FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(Ω↑↑7)
ψ011111111(0)))))))))
{n,n(1(1(1(1'1'2')2')2')2)2}
{n,n(1(1(1(1\41\42)2)2)2)2} = {n,n(1(1(1(1/1/2/22)2/22)2/22)2)2}
317FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(Ω↑↑8)
ψ0111111111(0))))))))))
{n,n(1(1(1(1(1'2')2')2')2')2)2}
{n,n(1(1(1(1\53)2)2)2)2} = {n,n(1(1(1(1(1/2/22)2/22)2/22)2/22)2)2}
318FGH ψ-function
FGH ψ(ψΩ2)
SAN
BAN
ψ(Ω↑↑9)
ψ01111111111(0)))))))))))
{n,n(1(1(1(1(1'1'2')2')2')2')2)2}
{n,n(1(1(1(1(1\51\52)2)2)2)2)2} = {n,n(1(1(1(1(1/1/2/22)2/22)2/22)2/22)2)2}
319name
proof theory
TON
FGH ψ-function

FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(cardinal)
FGH ψ(higher)

SAN (even)
SAN (odd)

BAN (even)
BAN (odd)
Bachmann-Howard ordinal
KPω
limit of C1(a,b) - system
ψ(ΩΩΩΩ...) = ψ(Ω↑↑ω) = ψ(εΩ+1) = ψ(φ(1,Ω+1))
ψ0111(...))))= ψ012)) = ψ(Ω2)
ψ012(0)))
ψ(2nd uncountable)
ψ(Ф(2)) = ψ(I(0,1)) = ψ(χ(1))
{n,n(1(1''2')2)2} = {n,n(1,,3)2} = {n,n(1...(1'1'2')...2')2} ~n/2-1 insert
{n,n(1(1''2')2)2} = {n,n(1,,3)2} = {n,n(1...(1'2')...2')2} ~(n+1)/2-1 insert
{n,n(1(1/23)2)2} = {n,n(1...(1\(n+2)/21\(n+2)/22)...2)2} ~n/2 insert = {n,n(1(1...(1/1/2/22)...2/22)2)2} ~n/2-1 insert
{n,n(1(1/23)2)2} = {n,n(1...(1\(n+3)/23)...2)2} ~(n+1)/2 insert = {n,n(1(1...(1/2/22)...2/22)2)2} ~(n+1)/2-1 insert
320FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)+1
ψ(Ω2)+1
ψ012(0)))+1
{n,n(2(1''2')2)2}
{n,n(2(1/23)2)2}
321FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)×2
ψ(Ω2)×2
ψ012(0)))+ψ012(0)))
{n,n(1(1(1''2')2)2(1''2')2)2}
{n,n(1(1(1/23)2)2(1/23)2)2}
322FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)×ω
ψ(Ω2+1)
ψ012(0))+ψ0(0))
{n,n(1(1(1''2')2)1,2(1''2')2)2}
{n,n(1(1(1/23)2)1,2(1/23)2)2}
323FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)ψ(εΩ+1)
ψ(Ω2+ψ(Ω2))
ψ012(0))+ψ012(0))))
{n,n(1(1(1(1''2')2)2(1''2')2)2(1''2')2)2}
{n,n(1(1(1(1/23)2)2(1/23)2)2(1/23)2)2}
324FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)ψ(εΩ+1)×ω
ψ(Ω2+ψ(Ω2+1))
ψ012(0))+ψ012(0))+1))
{n,n(1(2(1''2')2)2(1''2')2)2}
{n,n(1(2(1/23)2)2(1/23)2)2}
325FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)ψ(εΩ+1)ψ(εΩ+1)
ψ(Ω2+ψ(Ω2+ψ(Ω2)))
ψ012(0))+ψ012(0))+ψ012(0)))))
{n,n(1(1(1(1(1''2')2)2(1''2')2)2(1''2')2)2(1''2')2)2}
{n,n(1(1(1(1(1/23)2)2(1/23)2)2(1/23)2)2(1/23)2)2}
326FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1)ψ(εΩ+1)ψ(εΩ+1)...= εψ(εΩ+1)+1 = φ(1,ψ(εΩ+1)+1) = ψ(εΩ+1+Ω)
ψ(Ω2+Ω)
ψ012(0))+ψ1(0))
{n,n(1'2(1''2')2)2}
{n,n(1/2(1/23)2)2}
327FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ζψ(εΩ+1)+1 = φ(2,ψ(εΩ+1)+1) = ψ(εΩ+12)
ψ(Ω22)
ψ012(0))+ψ11(0)))
{n,n(1'1'2(1''2')2)2}
{n,n(1/1/2(1/23)2)2}
328FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
Гψ(εΩ+1)+1 = φ(1,0,ψ(εΩ+1)+1) = ψ(εΩ+1Ω)
ψ(Ω2Ω)
ψ012(0))+ψ111(0))))
{n,n(1(1'2')2(1''2')2)2}
{n,n(1(1/2/22)2(1/23)2)2}
329FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1ΩΩ)
ψ(Ω2ΩΩ)
ψ012(0))+ψ1111(0)))))
{n,n(1(1'1'2')2(1''2')2)2}
{n,n(1(1/1/2/22)2(1/23)2)2}
330FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1ΩΩΩ...) = ψ(εΩ+1×2)
ψ(Ω212))
ψ012(0))+ψ12(0)))
{n,n(1(1''2')3)2}
{n,n(1(1/23)3)2}
331FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×ω) = ψ(ωεΩ+1+1)
ψ(Ω212+1)) = ψ(Ω212)×ω) = ψ(Ω212)+ψ12)+...)
ψ012(0))+ψ12(0)+ψ0(0)))
{n,n(1(1''2')1,2)2}
{n,n(1(1/23)1,2)2}
332FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×ψ(εΩ+1))
ψ(Ω212+ψ(Ω2))
ψ012(0))+ψ12(0)+ψ012(0)))))
{n,n(1(1''2')1(1(1''2')2)2)2}
{n,n(1(1/23)1(1(1/23)2)2)2}
333FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×ψ(εΩ+1×...)) = ψ(εΩ+1×Ω) = ψ(ΩΩΩΩ...+1)
ψ(Ω212+Ω)) = ψ(Ω212+ψ(Ω2+ψ(Ω2+ψ(Ω2+...)))))
ψ012(0))+ψ12(0)+ψ1(0)))
{n,n(1(1''2')1'2)2}
{n,n(1(1/23)1/2)2}
334FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×Ωω)
ψ(Ω212ω))
ψ012(0))+ψ12(0)+ψ11(1))))
{n,n(1(1''2')1(2')2)2}
{n,n(1(1/23)1(2/22)2)2}
335FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×ΩΩ)
ψ(Ω212Ω))
ψ012(0))+ψ12(0)+ψ111(0)))))
{n,n(1(1''2')1(1'2')2)2}
{n,n(1(1/23)1(1/2/22)2)2}
336FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1×εΩ+1) = ψ(εΩ+12) = ψ(ωεΩ+1×2)
ψ(Ω21212)))
ψ012(0))+ψ12(0)+ψ12(0))))
{n,n(1(1''2')1(1''2')2)2}
{n,n(1(1/23)1(1/23)2)2}
337FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1ω) = ψ(ωεΩ+1×ω) = ψ(ωωεΩ+1+1)
ψ(Ω21212+1)))
ψ012(0))+ψ12(0)+ψ12(0)+ψ0(0))))
{n,n(1(2''2')2)2}
{n,n(1(2/23)2)2}
338FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1Ω) = ψ(ΩΩΩΩ...+1) = ψ(εΩ+1ψΩ+1ψΩ+1...))) = ψ(ωεΩ+1×ψ(ωεΩ+1×ψ(...))) = ψ(ωωεΩ+1)
ψ(Ω21212+Ω)))
ψ012(0))+ψ12(0)+ψ12(0)+ψ1(0))))
{n,n(1(1'2''2')2)2}
{n,n(1(1/2/23)2)2}
339FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1ΩΩΩ...) =ψ(εΩ+1εΩ+1) = ψ(ωωεΩ+1Ω+1) = ψ(ωωεΩ+1×2) = ψ(ωεΩ+12)
ψ(Ω2121212))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)))))
{n,n(1(1(1''2')2''2')2)2}
{n,n(1(1(1/23)2/23)2)2}
340FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1εΩ+1ω) = ψ(ωωωεΩ+1+1)
ψ(Ω2121212+1))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ0(0)))))
{n,n(1(1(1''2')1,2''2')2)2}
{n,n(1(1(1/23)1,2/23)2)2}
341FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1εΩ+1Ω) = ψ(ΩΩΩΩ...+1) = ψ(ωωεΩ+1×Ω)
ψ(Ω2121212+Ω))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ1(0)))))
{n,n(1(1(1''2')1'2''2')2)2}
{n,n(1(1(1/23)1/2/23)2)2}
342FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1εΩ+1εΩ+1) = ψ(ωωεΩ+12) = ψ(ωωωεΩ+1×2)
ψ(Ω212121212)))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0))))))
{n,n(1(1(1''2')1(1''2')2''2')2)2}
{n,n(1(1(1/23)1(1/23)2/23)2)2}
343FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1εΩ+1εΩ+1εΩ+1) = ψ(ωωωεΩ+12) = ψ(ωωωωεΩ+1×2)
ψ(Ω21212121212))))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)))))))
{n,n(1(1(1(1''2')2''2')2''2')2)2}
{n,n(1(1(1(1/23)2/23)2/23)2)2}
344FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+1εΩ+1εΩ+1εΩ+1εΩ+1) = ψ(ωωωωεΩ+12) = ψ(ωωωωωεΩ+1×2)
ψ(Ω2121212121212))))))))
ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0))))))))
{n,n(1(1(1(1''2')1(1''2')2''2')2''2')2)2}
{n,n(1(1(1(1/23)1(1/23)2/23)2/23)2)2}
345FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+2) = ψ(εΩ+1εΩ+1εΩ+1...) = ψ(ΩΩΩ...εΩ+1)
ψ(Ω22) = ψ(Ω2×2)
ψ012(0)+ψ2(0))) = ψ012(0))+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0)+ψ12(0))+...))))))
{n,n(1(1''3')2)2}
{n,n(1(1/24)2)2}
346FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+3) = ψ(εΩ+2εΩ+2εΩ+2...) = ψ(ΩΩΩ...εΩ+2)
ψ(Ω2×3)
ψ012(0)+ψ2(0)+ψ2(0)))
{n,n(1(1''4')2)2}
{n,n(1(1/25)2)2}
347FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ+ω)
ψ(Ω2×ω)
ψ012(1)) = ψ012(0)+ψ2(0)+ψ2(0)+...)
{n,n(1(1''1,2')2)2}
{n,n(1(1/21,2)2)2}
348FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ×2) = ψ(εΩ+ψ(εΩ+ψ(εΩ+...)))
ψ(Ω2×Ω)
ψ0121(0))))
{n,n(1(1''1'2')2)2}
{n,n(1(1/21/2)2)2}
349FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩ2) = ψ(εΩ×ψ(εΩ×ψ(εΩ×...)))
ψ(Ω2×Ω2)
ψ01211(0)))))
{n,n(1(1''1'1'2')2)2}
{n,n(1(1/21/1/2)2)2}
350FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εΩΩ)= ψ(εΩψ(εΩψ(εΩ...)))
ψ(Ω2×ΩΩ)
ψ012111(0))))))
{n,n(1(1''1(1'2')2')2)2}
{n,n(1(1/21(1/2/22)2)2)2}
351FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εεΩ+1) = ψ(εΩΩΩΩ...)
ψ(Ω2×ψ12))
ψ01212(0)))))
{n,n(1(1''1(1''2')2')2)2}
{n,n(1(1/21(1/23)2)2)2}
352FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(εεεΩ+1)
ψ(Ω2×ψ12×ψ12)))
ψ0121212(0)))))))
{n,n(1(1''1(1''1(1''2')2')2')2)2}
{n,n(1(1/21(1/21(1/23)2)2)2)2}
353FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ζΩ+1) = ψ(εεε...Ω+1) = ψ(φ(2,Ω+1))
ψ(Ω22)
ψ0122(0))))
{n,n(1(1''...2')2)2} ~n insert = {n,n(1(1''1''2')2)2}
{n,n(1(1/21...(1/23)...2)2)2} ~n-1 insert = {n,n(1(1/21/22)2)2}
354FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ζΩ+2) = ψ(εεε...ζΩ+1+1) = ψ(φ(2,Ω+2))
ψ(Ω22×2)
ψ0122(0))+ψ22(0))))
{n,n(1(1''1''3')2)2}
{n,n(1(1/21/23)2)2}
355FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ηΩ+1) = ψ(ζζζ...Ω+1) = ψ(φ(3,Ω+1))
ψ(Ω23)
ψ0122(0)+ψ2(0))))
{n,n(1(1''1''1''2')2)2}
{n,n(1(1/21/21/22)2)2}
356FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ηΩ+2) = ψ(ζζζ...ηΩ+1+1) = ψ(φ(3,Ω+2))
ψ(Ω23×2)
ψ0122(0)+ψ2(0))+ψ22(0)+ψ2(0))))
{n,n(1(1''1''1''3')2)2}
{n,n(1(1/21/21/23)2)2}
357FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(ω,Ω+1))
ψ(Ω2ω)
ψ0122(1))))
{n,n(1(1(2'')2')2)2} = {n,n(2,,3)2} = {n,n(1(1''2')2)2} ~n repeat
{n,n(1(1(2/32)2)2)2} ={n,n(1(1/22)2)2} ~n repeat for n >1
358FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(ψ(φ(ω,Ω+1)),Ω+1))
ψ(Ω2ψ(Ω2ω))
ψ01220122(1))))))))
{n,n(1(1(1(1(1(2'')2')2)2'')2')2)2}
{n,n(1(1(1(1(1(2/32)2)2)2/32)2)2)2}
359FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(ψ(φ(ψ(φ(ω,Ω+1)),Ω+1)),Ω+1))
ψ(Ω2ψ(Ω2ψ(Ω2ω)))
ψ012201220122(1))))))))))))
{n,n(1(1(1(1(1(1(1(1(2'')2')2)2'')2')2)2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1(2/32)2)2)2/32)2)2)2/32)2)2)2}
360FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(Ω,1)) = ψ(φ(Ω,Ω+1)) = ψ(φ(ψ(φ(ψ(φ(ψ(φ(...,Ω+1)),Ω+1)),Ω+1)),Ω+1))
ψ(Ω2Ω)
ψ01221(0)))))
{n,n(1(1(1'2'')2')2)2}
{n,n(1(1(1/2/32)2)2)2}
361FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(φ(Ω,1),0))
ψ(Ω2ψ12))
ψ012212(0))))))
{n,n(1(1(1(1(1'2'')2')2'')2')2)2}
{n,n(1(1(1(1(1/2/32)2)2/32)2)2)2}
362FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(φ(φ(Ω,1),0),0))
ψ(Ω2ψ12ψ12)))
ψ01221212(0))))))))
{n,n(1(1(1(1(1(1(1'2'')2')2'')2')2'')2')2)2}
{n,n(1(1(1(1(1(1(1/2/32)2)2/32)2)2/32)2)2)2}
363FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ГΩ+1) = ψ(φ(1,0,Ω+1)) = ψ(φ(φ(φ(...φ(Ω,1),...,0),0),0))
ψ(Ω2Ω2)
ψ01222(0)))))
{n,n(1(1(1''2'')2')2)2}
{n,n(1(1(1/22/32)2)2)2}
364FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(ГΩ+2) = ψ(φ(1,0,Ω+2))
ψ(Ω2Ω2×2)
ψ01222(0)))+ψ222(0)))))
{n,n(1(1(1''2'')2')3)2}
{n,n(1(1(1/22/32)3)2)2}
365FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,Ω+φ(1,0,Ω+1)))
ψ(Ω2Ω2×ψ(Ω2Ω2))
ψ01222(0))+ψ01222(0))))))))
{n,n(1(1(1''2'')2')1(1(1(1(1''2'')2')2)2)2)2}
{n,n(1(1(1/22/32)1(1(1(1(1/22/32)2)2)2)2)2)2}
366FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,Ω×2)) = ψ(φ(1,0,Ω+φ(1,0,Ω+φ(1,0,Ω+...))))
ψ(Ω2Ω2×Ω)
ψ01222(0))+ψ1(0))))
{n,n(1(1(1''2'')2')1(1'2)2)2}
{n,n(1(1(1/22/32)1/2)2)2}
367FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,φ(1,Ω+1)))
ψ(Ω2Ω2×ψ12))
ψ01222(0))+ψ12(0)))))
{n,n(1(1(1''2'')2')1(1''2')2)2}
{n,n(1(1(1/22/32)1(1/23)2)2)2}
368FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,φ(1,0,Ω+1)))
ψ(Ω2Ω2×ψ12Ω2))
ψ01222(0))+ψ1222(0)))))
{n,n(1(1(1''2'')2')1(1(1(1''2'')2')2)2)2}
{n,n(1(1(1/22/32)1(1(1(1/22/32)2)2)2)2)2}
369FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,1,Ω+1)) = ψ(φ(1,0,φ(1,0,φ(1,0,...φ(1,0,Ω+1)...))))
ψ(Ω2Ω2×Ω2) = ψ(Ω2Ω2+1)
ψ01222(0))+ψ2(0))))
{n,n(1(1(1''2'')2')1'2)2}
{n,n(1(1(1/22/32)1/22)2)2}
370FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,Ω,1)) = ψ(φ(1,φ(1,φ(1,...φ(1,0,Ω+1)...,0),0),0))
ψ(Ω2Ω2)
ψ01222(0))+ψ21(0)))))
{n,n(1(1(1''2'')2')1(1'2')2)2}
{n,n(1(1(1/22/32)1/2/22)2)2}
371FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,φ(1,Ω,1),0))
ψ(Ω2Ω212Ω2))
ψ01222(0))+ψ21222(0))))))
{n,n(1(1(1''2'')2')1(1(1''2'')2')2)2}
{n,n(1(1(1/22/32)1(1(1/22/32)2)2)2)2}
372FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,φ(1,φ(1,Ω,1),0),0))
ψ(Ω2Ω212Ω212Ω2)))
ψ01222(0))+ψ21222(0))+ψ21222(0)))))))))
{n,n(1(1(1''2'')1(1(1(1''2'')2')2'')2')2)2}
{n,n(1(1(1/22/32)1(1(1(1/22/32)2)2/32)2)2)2}
373FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,φ(1,φ(1,φ(1,Ω,1),0),0),0))
ψ(Ω2Ω212Ω212Ω212Ω2))))
ψ01222(0))+ψ21222(0))+ψ21222(0))+ψ21222(0))))))))))))
{n,n(1(1(1''2'')1(1(1(1''2'')1(1(1(1''2'')2')2'')2')2'')2')2)2}
{n,n(1(1(1/22/32)1(1(1(1(1(1/22/32)2)2/32)2)2/32)2)2)2}
374FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(2,0,Ω+1)) = ψ(φ(1,φ(1,φ(1,...φ(1,Ω,1)...,0),0),0))
ψ(Ω2Ω2×2)
ψ01222(0))+ψ22(0)))))
{n,n(1(1(1''2'')1(1''2'')2')2)2}
{n,n(1(1(1/22/32)1(1/22/32)2)2)2}
375FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(3,0,Ω+1))
ψ(Ω2Ω2×3)
ψ01222(0))+ψ22(0))+ψ22(0)))))
{n,n(1(1(1''2'')1(1''2'')1(1''2'')2')2)2}
{n,n(1(1(1/22/32)1(1/22/32)1(1/22/32)2)2)2}
376FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(ω,0,Ω+1))
ψ(Ω2Ω2×ω)
ψ01222(0)+ψ0(0)))))
{n,n(1(1(2''2'')2')2)2}
{n,n(1(1(2/22/32)2)2)2}
377FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(ψ(φ(ω,0,Ω+1)),0,Ω+1))
ψ(Ω2Ω2×ψ(Ω2Ω2))
ψ01222(0)+ψ01222(0))))))))))
{n,n(1(1(1(1(1(1''2'')2')2)2''2'')2')2)2}
{n,n(1(1(1(1(1(1/22/32)2)2)2/22/32)2)2)2}
378FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(Ω,0,1)) = ψ(φ(ψ(φ(ψ(φ(...,0,Ω+1)),0,Ω+1)),0,Ω+1))
ψ(Ω2Ω2×Ω)
ψ01222(0)+ψ1(0)))))
{n,n(1(1(1'2''2'')2')2)2}
{n,n(1(1(1/2/22/32)2)2)2}
379FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(φ(Ω,0,1),0,0))
ψ(Ω2Ω2×ψ12Ω2))
ψ01222(0)+ψ1222(0)))))))))
{n,n(1(1(1(1(1''2'')2')2''2'')2')2)2}
{n,n(1(1(1(1(1/22/32)2)2/22/32)2)2)2}
380FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(φ(φ(Ω,0,1),0,0),0,0))
ψ(Ω2Ω2×ψ12Ω2×ψ12Ω2)))
ψ01222(0)+ψ1222(0)+ψ1222(0)))))))))))))))
{n,n(1(1(1(1(1(1(1''2'')2')2''2'')2')2''2'')2')2)2}
{n,n(1(1(1(1(1(1(1/22/32)2)2/22/32)2)2/22/32)2)2)2}
381FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(φ(φ(φ(Ω,0,1),0,0),0,0),0,0))
ψ(Ω2Ω2×ψ12Ω2×ψ12Ω2×ψ12Ω2))))
ψ01222(0)+ψ1222(0)+ψ1222(0)+ψ1222(0))))))))))))))))))
{n,n(1(1(1(1(1(1(1(1(1''2'')2')2''2'')2')2''2'')2')2''2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1(1/22/32)2)2/22/32)2)2/22/32)2)2/22/32)2)2)2}
382FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,0,Ω+1)) = ψ(φ(φ(φ(...φ(Ω,0,1)...,0,0),0,0),0,0))
ψ(Ω2Ω22) = ψ(Ω2Ω2×Ω2)
ψ01222(0)+ψ2(0)))))
{n,n(1(1(1''3'')2')2)2}
{n,n(1(1(1/23/32)2)2)2}
383FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,0,0,Ω+1))
ψ(Ω2Ω23)
ψ01222(0)+ψ2(0)+ψ2(0)))))
{n,n(1(1(1''4'')2')2)2}
{n,n(1(1(1/24/32)2)2)2}
384FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(φ(1,0,...,Ω+1)) = φ(1ωΩ+10)
ψ(Ω2Ω2ω)
ψ01222(1)))))
{n,n(1(1(1''1,2'')2')2)2}
{n,n(1(1(1/21,2/32)2)2)2}
385FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1ΩΩ+10)
ψ(Ω2Ω2Ω) = ψ(Ω2Ω2ψ(Ω2Ω2...))
ψ012221(0))))))
{n,n(1(1(1''1'2'')2')2)2}
{n,n(1(1(1/21/2/32)2)2)2}
386FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1φ(Ω210))
ψ(Ω2Ω2ψ12Ω2)))
ψ012221222(0)))))))))
{n,n(1(1(1''1(1(1''2'')2')2'')2')2)2}
{n,n(1(1(1/21(1(1/22/32)2)2/32)2)2)2}
387FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1φ(1ωΩ+10))
ψ(Ω2Ω2ψ12Ω2ω)))
ψ012221222(1)))))))))
{n,n(1(1(1''1(1(1''1,2'')2')2'')2')2)2}
{n,n(1(1(1/21(1(1/21,2/32)2)2/32)2)2)2}
388FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1φ(1ΩΩ+10))
ψ(Ω2Ω2ψ12Ω2Ω)))
ψ0122212221(0))))))))))
{n,n(1(1(1''1(1(1''1'2'')2')2'')2')2)2}
{n,n(1(1(1/21(1(1/21/2/32)2)2/32)2)2)2}
389FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1φ(1φ(Ω210)))
ψ(Ω2Ω2ψ12Ω2ψ12Ω2))))
ψ0122212221222(0)))))))))))))
{n,n(1(1(1''1(1(1''1(1(1''2'')2')2'')2')2'')2')2)2}
{n,n(1(1(1/21(1(1/21(1(1/22/32)2)2/32)2)2/32)2)2)2}
390FGH ψΩ
FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
φ(1φ(1φ(1...φ(1ΩΩ+10)...)))
ψ(Ω2Ω2Ω2)
ψ012222(0))))))
{n,n(1(1(1''1''2'')2')2)2}
{n,n(1(1(1/21/22/32)2)2)2}
391FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω22)
ψ012222(0)))+ψ22(0)))))
{n,n(1(1(1''1''2'')1(1''2'')2')2)2}
{n,n(1(1(1/21/22/32)1(1/22/32)2)2)2}
392FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω222)
ψ012222(0)))+ψ222(0))))))
{n,n(1(1(1''1''2'')1(1''3'')2')2)2}
{n,n(1(1(1/21/22/32)1(1/23/32)2)2)2}
393FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2×2)
ψ012222(0)))+ψ2222(0)))))))
{n,n(1(1(1''1''2'')1(1''1''2'')2')2)2}
{n,n(1(1(1/21/22/32)1(1/21/22/32)2)2)2}
394FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2×ω)
ψ012222(0))+ψ0(0)))))
{n,n(1(1(2''1''2'')2')2)2}
{n,n(1(1(2/21/22/32)2)2)2}
395FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2×Ω2) = ψ(Ω2Ω2Ω2+1)
ψ012222(0))+ψ2(0)))))
{n,n(1(1(1''2''2'')2')2)2}
{n,n(1(1(1/22/22/32)2)2)2}
396FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2×2)
ψ012222(0))+ψ22(0))))))
{n,n(1(1(1''1''3'')2')2)2}
{n,n(1(1(1/21/23/32)2)2)2}
397FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω22)
ψ012222(0)+ψ2(0))))))
{n,n(1(1(1''1''1''2'')2')2)2}
{n,n(1(1(1/21/21/22/32)2)2)2}
398FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2ω)
ψ012222(1))))))
{n,n(1(1(1(2'')2'')2')2)2}
{n,n(1(1(1(2/32)2/32)2)2)2}
399FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2Ω)
ψ0122221(0)))))))
{n,n(1(1(1(1'2'')2'')2')2)2}
{n,n(1(1(1(1/2/32)2/32)2)2)2}
400FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2Ω2) = ψ(Ω2↑↑4)
ψ0122222(0)))))))
{n,n(1(1(1(1''2'')2'')2')2)2}
{n,n(1(1(1(1/22/32)2/32)2)2)2}
401FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2Ω2Ω2) = ψ(Ω2↑↑5)
ψ01222222(0))))))))
{n,n(1(1(1(1"1''2'')2'')2')2)2}
{n,n(1(1(1(1/21/22/32)2/32)2)2)2}
402FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2Ω2Ω2Ω2Ω2Ω2) = ψ(Ω2↑↑6)
ψ012222222(0)))))))))
{n,n(1(1(1(1(1"2'')2'')2'')2')2)2}
{n,n(1(1(1(1(1/22/32)2/32)2/32)2)2)2}
403FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2↑↑7)
ψ0122222222(0))))))))))
{n,n(1(1(1(1(1"1"2'')2'')2'')2')2)2}
{n,n(1(1(1(1(1/21/22/32)2/32)2/32)2)2)2}
404FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2↑↑8)
ψ01222222222(0)))))))))))
{n,n(1(1(1(1(1(1"2'')2'')2'')2'')2')2)2}
{n,n(1(1(1(1(1(1/22/32)2/32)2/32)2/32)2)2)2}
405FGH ψ(ψΩ2)
FGH ψ(ψΩ3)

SAN
BAN
ψ(Ω2↑↑9)
ψ012222222222(0))))))))))))
{n,n(1(1(1(1(1(1"1"2'')2'')2'')2'')2')2)2}
{n,n(1(1(1(1(1(1/21/22/32)2/32)2/32)2/32)2)2)2}
406FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)
FGH ψ(cardinal)
FGH ψ(higher)

SAN (even)
SAN (odd)

BAN (even)
BAN (odd)
ψ(Ω2↑↑ω) = ψ(εΩ2+1) = ψ(φ(1,Ω2+1))
ψ01222(...))))) = ψ0123))) = ψ(Ω3)
ψ0123(0))))
ψ(3d uncountable)
ψ(Ф(3)) = ψ(I(0,2)) = ψ(χ(2))
{n,n(1(1(1'''2'')2')2)2} = {n,n(1,,4)2} = {n,n(1(1...(1"1"2'')...2'')2')2} ~n/2-1 insert
{n,n(1(1(1'''2'')2')2)2} = {n,n(1,,4)2}= {n,n(1(1...(1"2'')...2'')2')2} ~(n+1)/2-1insert
{n,n(1(1(1/33)2)2)2} = {n,n(1(1(1...(1/21/22/32)...2/32)2)2} ~n/2-1 insert
{n,n(1(1(1/33)2)2)2} = {n,n(1(1(1...(1/22/32)...2/32)2)2)2} ~(n+1)/2-1 insert
407FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(εΩ2+2) = ψ(φ(1,Ω2+2))
ψ(Ω3×2)
ψ0123(0)+ψ3(0))))
{n,n(1(1(1'''3'')2')2)2}
{n,n(1(1(1/34)2)2)2}
408FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(ζΩ2+1) = ψ(φ(2,Ω2+1))
ψ(Ω32)
ψ01233(0)))))
{n,n(1(1(1'''1'''2'')2')2)2}
{n,n(1(1(1/31/32)2)2)2}
409FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(ζΩ2+2) = ψ(φ(2,Ω2+2))
ψ(Ω32×2)
ψ01233(0)+ψ33(0))))))
{n,n(1(1(1'''1'''3'')2')2)2}
{n,n(1(1(1/31/33)2)2)2}
410FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(ηΩ2+1) = ψ(φ(3,Ω2+1))
ψ(Ω33)
ψ01233(0)+ψ3(0)))))
{n,n(1(1(1'''1'''1'''2'')2')2)2}
{n,n(1(1(1/31/31/32)2)2)2}
411FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(ηΩ2+2) = ψ(φ(3,Ω2+2))
ψ(Ω33×2)
ψ01233(0)+ψ3(0)+ψ3(0)))))
{n,n(1(1(1'''1'''1'''3'')2')2)2}
{n,n(1(1(1/31/31/33)2)2)2}
412FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ω,Ω2+1))
ψ(Ω3ω)
ψ01233(1)))))
{n,n(1(1(1(2''')2'')2')2)2} = {n,n(2,,4)2}
{n,n(1(1(1(2/42)2)2)2)2}
413FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ψ(φ(ω,Ω2+1)),Ω2+1))
ψ(Ω3ψ(Ω3ω))
ψ0123301233(1))))))))))
{n,n(1(1(1(1(1(1(1(2''')2'')2')2)2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1(2/42)2)2)2)2/32)2)2)2)2}
414FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ψ(φ(ψ(φ(ω,Ω2+1)),Ω2+1)),Ω2+1))
ψ(Ω3ψ(Ω3ψ(Ω3ω)))
ψ012330123301233(1)))))))))))))))
{n,n(1(1(1(1(1(1(1(1(1(1(1(2''')2'')2')2)2''')2'')2')2)2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1(1(1(1(2/32)2)2)2)2/32)2)2)2)2/32)2)2)2)2}
415FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(Ω,Ω2+1)) = ψ(φ(ψ(φ(ψ(φ(ψ(φ(...,Ω2+1)),Ω2+1)),Ω2+1)),Ω2+1))
ψ(Ω3Ω)
ψ012331(0))))))
{n,n(1(1(1(1'2''')2'')2')2)2}
{n,n(1(1(1(1/2/42)2)2)2)2}
416FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ГΩ+12+1))
ψ(Ω3ψ12Ω2))
ψ012331222(0)))))))))
{n,n(1(1(1(1(1(1(1'2''')2'')2')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1/2/42)2)2)2/42)2)2)2)2}
417FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ψ1(φ(ГΩ+12+1)),Ω2+1))
ψ(Ω3ψ13ψ12Ω2)))
ψ0123312331222(0)))))))))))))
{n,n(1(1(1(1(1(1(1(1(1(1'2''')2'')2')2''')2'')2')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1(1(1/2/42)2)2)2/42)2)2)2/42)2)2)2)2}
418FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(ψ1(φ(ψ1(φ(ГΩ+12+1)),Ω2+1)),Ω2+1))
ψ(Ω3ψ13ψ13ψ12Ω2))))
ψ01233123312331222(0)))))))))))))))))
{n,n(1(1(1(1(1(1(1(1(1(1(1(1(1'2''')2'')2')2''')2'')2')2''')2'')2')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1(1(1(1(1(1/2/42)2)2)2/42)2)2)2/42)2)2)2/42)2)2)2)2}
419FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(Ω2,1)) = ψ(φ(Ω22+1)) = ψ(φ(ψ1(φ(ψ1(φ(ψ1(φ(...,Ω2+1)),Ω2+1)),Ω2+1)),Ω2+1))
ψ(Ω3Ω2)
ψ012332(0))))))
{n,n(1(1(1(1''2''')2'')2')2)2}
{n,n(1(1(1(1/22/42)2)2)2)2}
420FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(φ(Ω2,1),0))
ψ(Ω323))
ψ0123323(0)))))))
{n,n(1(1(1(1(1(1''2''')2'')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1/22/42)2)2/42)2)2)2)2}
421FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(φ(φ(φ(Ω2,1),0),0))
ψ(Ω323ψ23)))
ψ0123323323(0))))))))))
{n,n(1(1(1(1(1(1(1(1''2''')2'')2''')2'')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1(1(1/22/42)2)2/42)2)2/42)2)2)2)2}
422FGH ψ(ψΩ2)
FGH ψ(ψΩ3)
FGH ψ(ψΩ4)

SAN
BAN
ψ(ГΩ2+1) = ψ(φ(1,0,Ω2+1)) = ψ(φ(φ(φ(...φ(Ω2,1),...,0),0),0))
ψ(Ω3Ω3)
ψ012333(0))))))
{n,n(1(1(1(1'''2''')2'')2')2)2}
{n,n(1(1(1(1/32/42)2)2)2)2}
423FGH ψ(ψΩ3)
FGH ψ(ψΩ4)
FGH ψ(ψΩ5)
FGH ψ(cardinal)
FGH ψ(higher)

SAN
BAN
ψ(Ω3↑↑ω) = ψ(εΩ3+1) = ψ(φ(1,Ω3+1))
ψ(Ω4) = ψ01234)))) = ψ01233333(...))))))))
ψ01234(0)))))
ψ(4-th uncountable)
ψ(Ф(4)) = ψ(I(0,3)) = ψ(χ(3))
{n,n(1(1(1(1''''2''')2'')2')2)2} = {n,n(1,,5)2}
{n,n(1(1(1(1/43)2)2)2)2}
424FGH ψ(ψΩ3)
FGH ψ(ψΩ4)
FGH ψ(ψΩ5)

SAN
BAN
ψ(ГΩ3+1) = ψ(φ(1,0,Ω3+1))
ψ(Ω4Ω4)
ψ0123444(0)))))))
{n,n(1(1(1(1(1''''2'''')2''')2'')2')2)2}
{n,n(1(1(1(1(1/51/42)2)2)2)2)2}
425FGH ψ(ψΩ4)
FGH ψ(ψΩ5)
FGH ψ(ψΩ6)
FGH ψ(cardinal)

FGH ψ(higher)
SAN
BAN
ψ(Ω4↑↑ω) = ψ(εΩ4+1) = ψ(φ(1,Ω4+1))
ψ(Ω5) = ψ012345))))) = ψ012344444(...)))))))))
ψ012345(0))))))
ψ(5-th uncountable)
ψ(Ф(5)) = ψ(I(0,4)) = ψ(χ(4))
{n,n(1(1(1(1(1'''''2'''')2''')2'')2')2)2} = {n,n(1,,6)2}
{n,n(1(1(1(1(1/53)2)2)2)2)2}
426FGH ψ(ψΩ4)
FGH ψ(ψΩ5)
FGH ψ(ψΩ6)

SAN
BAN
ψ(ГΩ4+1) = ψ(φ(1,0,Ω4+1))
ψ(Ω5Ω5)
ψ01234555(0))))))))
{n,n(1(1(1(1(1(1'''''2''''')2'''')2''')2'')2')2)2}
{n,n(1(1(1(1(1(1/61/52)2)2)2)2)2)2}
427name
proof theory
FGH ψ-function
FGH ψ(cardinal)

FGH ψ(higher)
SAN
BAN
Buchholz ordinal
П11-CA0
ψ(Ωω) = ψ(Ωsup(n)) = ψ0123456(...)))))))
ψ(ω-th uncountable)
ψ(Ф(ω)) = ψ(I(0,ω)) = ψ(χ(sup(n)))
{n,n(1,,1,2)2} = {n,n(1(1(1(1(1(1(...)2''''')2'''')2''')2'')2')2)2}
{n,n(1(2//2)2)2} = {n,n(1(2/1,22)2)2} = {n,n(1...(1/n3)...2)2} ~n insert
428FGH ψ-function
SAN
BAN
ψ(Ωω)+1
{n,n(2,,1,2)2} = {n,n(2(1,,1,2)2)2}
{n,n(2(2//2)2)2}
429FGH ψ-function
SAN
BAN
ψ(Ωω)+ω
{n,n(1,2,,1,2)2} = {n,n(1,2(1,,1,2)2)2}
{n,n(1,2(2//2)2)2}
430FGH ψ-function
SAN
BAN
ψ(Ωω)+ε0
{n,n(1(1(1,,2)2)2(1,,1,2)2)2}
{n,n(1(1/2)2(2//2)2)2}
431FGH ψ-function
SAN
BAN
ψ(Ωω)×2
{n,n(1(1(1,,1,2)2)2(1,,1,2)2)2}
{n,n(1(1(2//2)2)2(2//2)2)2}
432FGH ψ-function
SAN
BAN
ψ(Ωω)×ω = ψ(Ωω+1)
{n,n(1(1(1,,1,2)2)1,2(1,,1,2) 2)2}
{n,n(1(1(2//2)2)1,2(2//2)2)2}
433FGH ψ-function
SAN
BAN
ψ(Ωω)ψ(Ωω) = ψ(Ωω+ψ(Ωω))
{n,n(1(1(1(1,,1,2)2)2(1,,1,2)2)2(1,,1,2)2)2}
{n,n(1(1(1(2//2)2)2(2//2)2)2(2//2)2)2}
434FGH ψ-function
SAN
BAN
ψ(Ωω)ψ(Ωω)ψ(Ωω) = ψ(Ωω+ψ(Ωω+ψ(Ωω)))
{n,n(1(1(1(1(1,,1,2)2)2(1,,1,2)2)2(1,,1,2)2)2(1,,1,2)2)2}
{n,n(1(1(1(1(2//2)2)2(2//2)2)2(2//2)2)2(2//2)2)2}
435FGH ψ-function
SAN
BAN
εψ(Ωω)+1 = φ(1,ψ(Ωω)+1) = ψ(Ωω+Ω) = ψ(Ωω+ψ(Ωω+ψ(Ωω+...)))
{n,n(1'2(1,,1,2)2) = {n,n(1(1,,2)2(1,,1,2)2)2}
{n,n(1/2(2//2)2)2}
436FGH ψ-function
SAN
BAN
ζψ(Ωω)+1 = φ(2,ψ(Ωω)+1) = ψ(Ωω2)
{n,n(1'1'2(1,,1,2)2) = {n,n(1(1,,2)1(1,,2)2(1,,1,2)2)2}
{n,n(1/1/2(2//2)2)2}
437FGH ψ-function
SAN
BAN
ηψ(Ωω)+1 = φ(3,ψ(Ωω)+1) = ψ(Ωω3)
{n,n(1'1'1'2(1,,1,2)2) = {n,n(1(1,,2)1(1,,2)1(1,,2)2(1,,1,2)2)2}
{n,n(1/1/1/2(2//2)2)2}
438FGH ψ-function
SAN
BAN
φ(ω,ψ(Ωω)+1) = ψ(Ωωω)
{n,n(1(2,,2)2(1,,1,2)2)2}
{n,n(1(2/22)2(2//2)2)2}
439FGH ψ-function
SAN
BAN
Гψ(Ωω)+1 =  φ(1,0,ψ(Ωω)+1) = ψ(ΩωΩ)
{n,n(1(1(1,,2)2,,2)2(1,,1,2)2)2}
{n,n(1(1/2/22)2(2//2)2)2}
440FGH ψ-function
SAN
BAN
φ(1,0,0,ψ(Ωω)+1) = ψ(ΩωΩ2)
{n,n(1(1(1,,2)3,,2)2(1,,1,2)2)2}
{n,n(1(1/3/22)2(2//2)2)2}
441FGH ψ-function
SAN
BAN
ψ(ΩωΩ+1) = ψ(Ωω12))
{n,n(1(1,,3)2(1,,1,2)2)2} = {n,n(1''2(1,,1,2)2)2}
{n,n(1(1/23)2(2//2)2)2}
442FGH ψ-function
SAN
BAN
ψ(Ωω1Ω2+1)) = ψ(Ωω13))
{n,n(1(1,,4)2(1,,1,2)2)2} = {n,n(1'''2(1,,1,2)2)2}
{n,n(1(1(1(2//22)3)2(2//22)2)2)2}
443FGH ψ-function
SAN
BAN
ψ(Ωω12Ω3+1))) = ψ(Ωω14))
{n,n(1(1,,5)2(1,,1,2)2)2} = {n,n(1''''2(1,,1,2)2)2}
{n,n(1(1(1(1(2//23)3)2(2//23)2)2)2)2}
444FGH ψ-function
SAN
BAN
ψ(Ωω123Ω4+1))) = ψ(Ωω15))
{n,n(1(1,,6)2(1,,1,2)2)2} = {n,n(1'''''2(1,,1,2)2)2}
{n,n(1(1(1(1(1(2//24)3)2(2//24)2)2)2)2)2}
445FGH ψ-function
SAN
BAN
ψ(Ωω1ω)) = ψ(Ωω+ ψ12345(...))))))
{n,n(1(1,,1,2)3)2}
{n,n(1(3//2)2)2}
446FGH ψ-function
SAN
BAN
ψ(Ωω1ω)×2)
{n,n(1(1,,1,2)4)2}
{n,n(1(3//2)2)2}
447FGH ψ-function
SAN
BAN
ψ(Ωω1ω)×ω) = ψ(Ωω+ ψ1ω+1))
{n,n(1(1,,1,2)1,2)2}
{n,n(1(1,2//2)2)2}
448FGH ψ-function
SAN
BAN
ψ(Ωωψ1ω)+1) = ψ(Ωω+ ψ1ω+Ω))
{n,n(1(1,,1,2)1'2)2}
{n,n(1(1//3)2)2}
449FGH ψ-function
SAN
BAN
ψ(Ωω1ω1ω))) = ψ(Ωω1ω12345(...)))))))
{n,n(1(1,,1,2)1(1,,1,2)2)2}
{n,n(1(1//1(2//2)2)2)2}
450FGH ψ-function
SAN
BAN
ψ(Ωω1ω1ω+Ω)))
{n,n(1(1,,2)2,,1,2)2} = {n,n(1'2,,1,2)2}
{n,n(1(1//1(2//2)1(1/2/22)2)2)2}
451FGH ψ-function
SAN
BAN
ψ(Ωω1ω1ω1ω))))
{n,n(1(1(1(1,,1,2)2,,2)2(1,,1,2)2,,2)2)2}
{n,n(1(1//1(2//2)1(2//2)2)2)2}
452FGH ψ-function
SAN
BAN
ψ(Ωω1ω1ω1ω1(...))))) = ψ(Ωω2)
{n,n(1(1''2(1,,1,2)2,,2)2)2} = {n,n(1(1,,3)2(1,,1,2)2,,2)2}
{n,n(1(1//1(1/22(2//22)2)2)2)2}
453FGH ψ-function
SAN
BAN
ψ(ΩωΩ2+1) = ψ(Ωω23))
{n,n(1(1'''2(1,,1,2)2,,2)2)2} = {n,n(1(1,,4)2(1,,1,2)2,,2)2}
{n,n(1(1//1(1(1/33)2(2//22)2)2)2)2}
454FGH ψ-function
SAN
BAN
ψ(Ωω2Ω3+1)) = ψ(Ωω24))
{n,n(1(1''''2(1,,1,2)2,,2)2)2} = {n,n(1(1,,5)2(1,,1,2)2,,2)2}
{n,n(1(1//1(1(1(1/43)2)2(2//22)2)2)2)2}
455FGH ψ-function
SAN
BAN
ψ(Ωω23Ω4+1))) = ψ(Ωω25))
{n,n(1(1'''''2(1,,1,2)2,,2)2)2} = {n,n(1(1,,6)2(1,,1,2)2,,2)2}
{n,n(1(1//1(1(1(1(1/53)2)2)2(2//22)2)2)2)2}
456FGH ψ-function
SAN
BAN
ψ(Ωω2ω)) = ψ(Ωω2345(...)))))
{n,n(1(1(1,,1,2)3,,2)2)2}
{n,n(1(1//1(1(2//22)3)2)2)2}
457FGH ψ-function
SAN
BAN
ψ(Ωω2ω)×2)
{n,n(1(1(1,,1,2)4,,2)2)2}
{n,n(1(1//1(1(2//22)4)2)2)2}
458FGH ψ-function
SAN
BAN
ψ(Ωω2ω)×ω) = ψ(Ωω2ω+1))
{n,n(1(1(1,,1,2)1,2,,2)2)2}
{n,n(1(1//1(1(2//22)1,2)2)2)2}
459FGH ψ-function
SAN
BAN
ψ(Ωω2ω2ω)))
{n,n(1(1(1,,1,2)1(1,,1,2)2,,2)2)2}
{n,n(1(1//1(2//22)2)2)2}
460FGH ψ-function
SAN
BAN
ψ(Ωω2ω2ω2)))
{n,n(1(1,,3)2,,1,2)2} = {n,n(1''2,,1,2)2}
{n,n(1(1//1(1(2//22)1(1/22/32)22)2)2)2}
461FGH ψ-function
SAN
BAN
ψ(Ωω2ω2ω2ω))))
{n,n(1(1(1(1(1,,1,2)2,,2)2(1,,1,2)2,,2)2,,2)2)2}
{n,n(1(1//1(1(2//22)1(2//22)22)2)2)2}
462FGH ψ-function
SAN
BAN
ψ(Ωω2ω2ω2ω2(...))))) = ψ(Ωω3)
{n,n(1(1(1,,3)2(1,,1,2)2,,2)2,,2)2}
{n,n(1(1//1(1(1/32(2//23)2)22)2)2)2}
463FGH ψ-function
SAN
BAN
ψ(Ωω3ω)) = ψ(Ωω345(...))))
{n,n(1(1(1(1,,1,2)3,,2)2,,2)2)2}
{n,n(1(1//1(1(1(2//23)3)22)2)2)2}
464FGH ψ-function
SAN
BAN
ψ(Ωω3ω3ω)))
{n,n(1(1(1(1,,1,2)1(1,,1,2)2,,2)2,,2)2)2}
{n,n(1(1//1(1//1(2//23)22)2)2)2}
465FGH ψ-function
SAN
BAN
ψ(Ωω+ ψ3ω3ω3)))
{n,n(1(1,,4)2,,1,2)2} = {n,n(1'''2,,1,2)2}
{n,n(1(1//1(1//1(1(2//23)1(1/32/42)23)22)2)2)2}
466FGH ψ-function
SAN
BAN
ψ(Ωω3ω3ω3ω3(...))))) = ψ(Ωω4)
{n,n(1(1(1(1,,3)2(1,,1,2)2,,2)2,,2)2,,2)2}
{n,n(1(1//1(1//1(1(1/42(2//24)2)23)22)2)2)2}
467FGH ψ-function
SAN
BAN
ψ(Ωω4ω)) = ψ(Ωω45(...)))
{n,n(1(1(1(1(1,,1,2)3,,2)2,,2)2,,2)2)2}
{n,n(1(1//1(1(1(1(2//24)3)23)22)2)2)2}
468FGH ψ-function
SAN
BAN
ψ(Ωω4ω4ω)))
{n,n(1(1(1(1(1,,1,2)1(1,,1,2)2,,2)2,,2)2,,2)2)2}
{n,n(1(1//1(1//1(1//1(2//24)23)22)2)2)2}
469FGH ψ-function
SAN
BAN
ψ(Ωω4ω4ω4)))
{n,n(1(1,,5)2,,1,2)2} = {n,n(1''''2,,1,2)2}
{n,n(1(1//1(1//1(1//1(1(2//24)1(1/42/52)24)23)22)2)2)2}
470FGH ψ-function
SAN
BAN
ψ(Ωωω) = ψ(Ωω×2)
{n,n(1(1,,1,2)2,,1,2)2}
{n,n(1(1//1//2)2)2}
471FGH ψ-function
SAN
BAN
ψ(Ωω×3)
{n,n(1(1,,1,2)3,,1,2)2}
{n,n(1(1//1//3)2)2}
472FGH ψ-function
SAN
BAN
ψ(Ωω×ω)
{n,n(1(1,,1,2)1,2,,1,2)2}
{n,n(1(1//1//1,2)2)2}
473FGH ψ-function
SAN
BAN
ψ(Ωω×Ω) = ψ(Ωω×ψ(Ωω×ψ(Ωω×...)))
{n,n(1(1,,1,2)1'2,,1,2)2}
{n,n(1(1//1//1/2)2)2}
474FGH ψ-function
SAN
BAN
ψ(Ωω×ψ12))
{n,n(1(1,,1,2)1(1(1,,3)2,,2)2,,1,2)2}
{n,n(1(1//1//1(1/23)2(2//2)2)2)2}
475FGH ψ-function
SAN
BAN
ψ(Ωω×ψ1ω))
{n,n(1(1,,1,2)1(1(1,,1,2)2,,2)2,,1,2)2}
{n,n(1(1//1//1(3//2)2)2)2}
476FGH ψ-function
SAN
BAN
ψ(Ωω×ψ1ω×ψ1ω)))
{n,n(1(1,,1,2)1(1(1(1,,1,2)2,,2)2,,1,2)2,,2)2,,1,2)2}
{n,n(1(1//1//1(1//1//1(3//2)2)2)2)2}
477FGH ψ-function
SAN
BAN
ψ(Ωω×Ω2) = ψ(Ωω×ψ1ω×ψ1ω×...)))
{n,n(1(1,,1,2)1''2,,1,2)2}
{n,n(1(1//1//1/22)2)2}
478FGH ψ-function
SAN
BAN
ψ(Ωω×Ω3) = ψ(Ωω×ψ2ω×ψ2ω×...)))
{n,n(1(1,,1,2)1'''2,,1,2)2}
{n,n(1(1//1//1(1//1//1/322)2)2)2}
479FGH ψ-function
SAN
BAN
ψ(Ωω2) = ψ(Ωω×Ωω)
{n,n(1(1,,1,2)1(1,,1,2)2,,1,2)2}
{n,n(1(1//1//1//2)2)2}
480FGH ψ-function
SAN
BAN
ψ(Ωω3)
{n,n(1(1,,1,2)1(1,,1,2)1(1,,1,2)2,,1,2)2}
{n,n(1(1//1//1//1//2)2)2}
481FGH ψ-function
SAN
BAN
ψ(Ωωω) = ψ(Ωω×Ωω)
{n,n(1(2,,1,2)2,,1,2)2}
{n,n(1(1(2//22)2)2)2} = {n,n(1(1(2/2,22)2)2)2}
482FGH ψ-function
SAN
BAN
ψ(ΩωΩ) = ψ(Ωωψ(Ωωψ(Ωω...)))
{n,n(1(1'2,,1,2)2,,1,2)2}
{n,n(1(1(1/2//22)2)2)2}
483FGH ψ-function
SAN
BAN
ψ(Ωωψ12))
{n,n(1(1(1(1,,3)2,,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1(1/23)2//22)2)2)2}
484FGH ψ-function
SAN
BAN
ψ(Ωωψ1ω))
{n,n(1(1(1(1,,1,2)2,,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1(2//2)2//22)2)2)2}
485FGH ψ-function
SAN
BAN
ψ(Ωωψ1ωψ1ω)))
{n,n(1(1(1(1(1(1,,1,2)2,,2)2,,1,2)2,,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1(1(1(2//2)2//22)2)2//22)2)2)2}
486FGH ψ-function
SAN
BAN
ψ(ΩωΩ2) = ψ(Ωωψ1ωψ1ω...)))
{n,n(1(1(1,,3)2,,1,2)2,,1,2)2}
{n,n(1(1(1/22//22)2)2)2}
487FGH ψ-function
SAN
BAN
ψ(ΩωΩ3) = ψ(Ωωψ2ωψ2ω...)))
{n,n(1(1(1,,4)2,,1,2)2,,1,2)2}
{n,n(1(1(1(1(1/32//22)22)2//22)2)2)2}
488FGH ψ-function
SAN
BAN
ψ(ΩωΩω)
{n,n(1(1(1,,1,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1//2//22)2)2)2}
489FGH ψ-function
SAN
BAN
ψ(ΩωΩωΩω)
{n,n(1(1(1,,1,2)1(1,,1,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1//1//2//22)2)2)2}
490FGH ψ-function
SAN
BAN
ψ(ΩωΩωΩωΩω)
{n,n(1(1(1(1,,1,2)2,,1,2)2,,1,2)2,,1,2)2}
{n,n(1(1(1(1//2//22)2//22)2)2)2}
491name
proof theory
FGH ψ-function

FGH ψ(higher)
SAN
BAN
Takeuti-Feferman-Buchholz ordinal
П11-CA0+BI
ψ(ΩωΩωΩωΩω...) = ψ(εΩω+1) = ψ(φ(1,Ωω+1)) = ψ(Ωω+1)
ψ(Ф(ω+1)) = ψ(I(0,ω+1)) = ψ(χ(ω))
{n,n(1(1,,2,2)2}
{n,n(1(1(1//23)2)2)2} = {n,n(1(1(1/2,23)2)2)2}
492FGH ψ-function
SAN
BAN
ψ(Ωω+12) = ψ(ζΩω+1) = ψ(φ(2,Ωω+1))
{n,n(1(1,,2,2)1(1,,2,2)2,,1,2)2}
{n,n(1(1(1//21//22)2)2)2}
493FGH ψ-function
SAN
BAN
ψ(Ωω+13) = ψ(ηΩω+1) = ψ(φ(3,Ωω+1))
{n,n(1(1,,2,2)1(1,,2,2)1(1,,2,2)2,,1,2)2}
{n,n(1(1(1//21//21//22)2)2)2}
494FGH ψ-function
SAN
BAN
ψ(Ωω+1ω) = ψ(φ(ω,Ωω+1))
{n,n(1(2,,2,2)2,,1,2)2}
{n,n(1(1(1(2//32)2)2)2)2} = {n,n(1(1(1(2/3,22)2)2)2)2}
495FGH ψ-function
SAN
BAN
ψ(Ωω+1Ω) = ψ(φ(Ω,Ωω+1))
{n,n(1(1'2,,2,2)2,,1,2)2}
{n,n(1(1(1(1/2//32)2)2)2)2}
496FGH ψ-function
SAN
BAN
ψ(Ωω+1Ωω) = ψ(φ(Ωωω+1)) = ψ(φ(Ωω,1))
{n,n(1(1(1,,1,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1//2//32)2)2)2)2}
497FGH ψ-function
SAN
BAN
ψ(Ωω+1ψωω+1)) = ψ(φ(φ(Ωω,1),0))
{n,n(1(1(1,,2,2)2,,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1(1//23)2//32)2)2)2)2}
498FGH ψ-function
SAN
BAN
 ψ(Ωω+1ψωω+1ψωω+1))) = ψ(φ(φ(φ(Ωω,1),0),0))
{n,n(1(1(1(1,,2,2)2,,2)2,,2,2)2,,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1(1(1(1//23)2//32)2)2//32)2)2)2)2}
499FGH ψ-function
SAN
BAN
ψ(Ωω+1ψωω+1ψωω+1ψωω+1)))) = ψ(φ(φ(φ(φ(Ωω,1),0),0),0))
{n,n(1(1(1(1(1(1,,2,2)2,,2)2,,2,2)2,,2)2,,2,2)2,,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1(1(1(1(1(1//23)2//32)2)2//32)2)2//32)2)2)2)2}
500FGH ψ-function
SAN
BAN
ψ(Ωω+1Ωω+1) = ψ(Ωω+1ψωω+1ψωω+1ψωω+1)...))) = ψ(ГΩω+1) = ψ(φ(φ(φ(...φ(Ωω,1),...,0),0),0))
{n,n(1(1(1,,2,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1//22//32)2)2)2)2}
501FGH ψ-function
SAN
BAN
ψ(Ωω+1Ωω+1Ωω+1)
{n,n(1(1(1,,2,2)1(1,,2,2)2,,2,2)2,,1,2)2}
{n,n(1(1(1(1//21//22//32)2)2)2)2}
502FGH ψ-function
FGH ψ(higher)

SAN
BAN
ψ(εΩω+1+1) = ψ(Ωω+2)
ψ(Ф(ω+2)) = ψ(I(0,ω+2)) = ψ(χ(ω+1))
{n,n(1,,3,2)2}
{n,n(1(1(1(1//33)2)2)2)2} = {n,n(1(1(1(1/3,23)2)2)2)2}
503FGH ψ-function
SAN
BAN
ψ(ГΩω+1+1) = ψ(Ωω+2Ωω+2)
{n,n(1(1(1,,3,2)2,,3,2)2,,1,2)2}
{n,n(1(1(1(1(1//32//42)2)2)2)2)2}
504FGH ψ-function
SAN
BAN
ψ(εΩω+2+1) = ψ(Ωω+3)
ψ(Ф(ω+3)) = ψ(I(0,ω+3)) = ψ(χ(ω+2))
{n,n(1,,4,2)2}
{n,n(1(1(1(1(1//43)2)2)2)2)2} = {n,n(1(1(1(1(1/4,23)2)2)2)2)2}
505FGH ψ-function
SAN
BAN
ψ(ГΩω+2+1) = ψ(Ωω+3Ωω+3)
{n,n(1(1(1,,4,2)2,,4,2)2,,1,2)2}
{n,n(1(1(1(1(1(1//42//52)2)2)2)2)2)2}
506FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω×2) = ψ(Ωω+sup(n))
ψ(Ф(ω×2)) = ψ(I(0,ω×2)) = ψ(χ(ω+sup(n)))
{n,n(1,,1,3)2}
{n,n(1(1(2///2)2)2)2} = {n,n(1(1(2/1,32)2)2)2}
507FGH ψ-function
SAN
BAN
ψ(Ωω×21ω×21ω×2)))
{n,n(1(1,,1,3)1(1,,1,3)2)2}
{n,n(1(1(1///1(2//2)2)2)2)2}
508FGH ψ-function
SAN
BAN
ψ(Ωω×22ω×22ω×2)))
{n,n(1(1(1,,1,3)1(1,,1,3)2,,2)2)2}
{n,n(1(1(1///1(2//22)2)2)2)2}
509FGH ψ-function
SAN
BAN
ψ(Ωω×2ωω×2ωω×2)))
{n,n(1(1(1,,1,3)1(1,,1,3)2,,1,2)2)2}
{n,n(1(1(1///1(2///2)2)2)2)2}
510FGH ψ-function
SAN
BAN
ψ(Ωω×2ω+1ω×2ω+1ω×2)))
{n,n(1(1(1(1,,1,3)1(1,,1,3)2,,1,2)2,,1,2)2)2}
{n,n(1(1(1///1(2///22)2)2)2)2}
511FGH ψ-function
SAN
BAN
ψ(Ωω×2×2)
{n,n(1(1,,1,3)2,,1,3)2}
{n,n(1(1(1///1///2)2)2)2}
512FGH ψ-function
SAN
BAN
ψ(Ωω×2ω)
{n,n(1(2,,1,3)2,,1,3)2}
{n,n(1(1(1(2///22)2)2)2)2}
513FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(εΩω×2+1) = ψ(Ωω×2+1)
ψ(Ф(ω×2+1)) = ψ(I(0,ω×2+1)) = ψ(χ(ω×2))
{n,n(1,,2,3)2}
{n,n(1(1(1(1///23)2)2)2)2}
514FGH ψ-function
SAN
BAN
ψ(ГΩω×2+1) = ψ(Ωω×2+1Ωω×2+1)
{n,n(1(1(1,,2,3)2,,2,3)2,,1,3)2}
{n,n(1(1(1(1(1///21///32)2)2)2)2)2}
515FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(εΩω×2+1+1) = ψ(Ωω×2+2)
ψ(Ф(ω×2+2)) = ψ(I(0,ω×2+2)) = ψ(χ(ω×2+1))
{n,n(1,,3,3)2}
{n,n(1(1(1(1(1///33)2)2)2)2)2}
516FGH ψ-function
SAN
BAN
ψ(ГΩω×2+1+1) = ψ(Ωω×2+2Ωω×2+2)
{n,n(1(1(1,,3,3)2,,3,3)2,,1,3)2}
{n,n(1(1(1(1(1(1///31///42)2)2)2)2)2)2}
517FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω×3)
ψ(Ф(ω×3)) = ψ(I(0,ω×3)) = ψ(χ(ω×2+sup(n)))
{n,n(1,,1,4)2}
{n,n(1(1(1(2////2)2)2)2)2} = {n,n(1(1(1(2/1,42)2)2)2)2}
518FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω×4)
ψ(Ф(ω×4)) = ψ(I(0,ω×4)) = ψ(χ(ω×3+sup(n)))
{n,n(1,,1,5)2}
{n,n(1(1(1(1(2/////2)2)2)2)2)2} = {n,n(1(1(1(1(2/1,52)2)2)2)2)2}
519FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω×5)
ψ(Ф(ω×5)) = ψ(I(0,ω×5)) = ψ(χ(ω×4+sup(n)))
{n,n(1,,1,6)2}
{n,n(1(1(1(1(1(2//////2)2)2)2)2)2)2} = {n,n(1(1(1(1(1(2/1,62)2)2)2)2)2)2}
520FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω2) = ψ(sup(Ωω×n))
ψ(Ф(ω2)) = ψ(I(0,ω2))
{n,n(1,,1,1,2)2}
{n,n(1(2/1,1,22)2)2} = {n,n(1...(2//2)...2)2} ~n-insert and repeat
521FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω2)
ψ(Ф(ω2+ω)) = ψ(I(0,ω2+ω))
{n,n(1,,1,2,2)2}
{n,n(1(1(2/1,2,22)2)2)2}
522FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω2×2)
ψ(Ф(ω2×2)) = ψ(I(0,ω2×2))
{n,n(1,,1,1,3)2}
{n,n(1(1(2/1,1,32)2)2)2}
523FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω2×3)
ψ(Ф(ω2×3)) = ψ(I(0,ω2×3))
{n,n(1,,1,1,4)2}
{n,n(1(1(1(2/1,1,42)2)2)2)2}
524FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω2×4)
ψ(Ф(ω2×4)) = ψ(I(0,ω2×4))
{n,n(1,,1,1,5)2}
{n,n(1(1(1(1(2/1,1,52)2)2)2)2)2}
525FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω3)
ψ(Ф(ω3)) = ψ(I(0,ω3))
{n,n(1,,1,1,1,2)2}
{n,n(1(2/1,1,1,22)2)2}
526FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωω4)
ψ(Ф(ω4)) = ψ(I(0,ω4))
{n,n(1,,1,1,1,1,2)2}
{n,n(1(2/1,1,1,1,22)2)2}
527FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωωω)
ψ(Ф(ωω)) = ψ(I(0,ωω))
{n,n(1,,1(2)2)2}
{n,n(1(2/1(2)22)2)2}
528FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωωωω)
ψ(Ф(ωωω)) = ψ(I(0,ωωω))
{n,n(1,,1(1,2)2)2}
{n,n(1(2/1(1,2)22)2)2}
529FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωωωωω)
ψ(Ф(ωωωω)) = ψ(I(0,ωωωω))
{n,n(1,,1(1(1)2)2)2}
{n,n(1(2/1(1(2)2)22)2)2}
530FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωωωω...) = ψ(Ωε0) = ψ(Ωψ(Ω))
ψ(Ф(ωωω...)) = ψ(Ф(ψ(Ф(1)))) = ψ(I(0,ωωω...)) = ψ(I(0,ψ(I(0,0))))
{n,n(1,,1(1'2)2)2}
{n,n(1(2/1(1/2)22)2)2}
531FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(ΩГ0) = ψ(Ωψ(ΩΩ))
ψ(Ф(ψ(Ф(1)Ф(1)))) = ψ(I(0,ψ(I(0,0)I(0,0))))
{n,n(1,,1(1(1'2')2)2)2}
{n,n(1(2/1(1(1/2/22)2)22)2)2}
532FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(ΩΩ2))
ψ(Ф(ψ(Ф(1)Ф(1)2))) = ψ(I(0,ψ(I(0,0)I(0,0)2)))
{n,n(1,,1(1(1'3')2)2)2}
{n,n(1(2/1(1(1/3/22)2)22)2)2}
533FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(ΩΩΩ))
ψ(Ф(ψ(Ф(1)Ф(1)Ф(1)))) = ψ(I(0,I(0,0)I(0,0)I(0,0))))
{n,n(1,,1(1(1'1'2')2)2)2}
{n,n(1(2/1(1(1/1/1/22)2)22)2)2}
534FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(εΩ+1)) = ψ(Ωψ(Ω2))
ψ(Ф(ψ(Ф(2)))) = ψ(I(0,ψ(I(0,1))))
{n,n(1,,1(1(1(1''2')2)2)2)2} = {n,n(1,,1(1(1,,3)2)2)2}
{n,n(1(2/1(1(1(1/22/32)2)2)22)2)2}
535FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(ГΩ+1)) = ψ(Ωψ(Ω2Ω2))
ψ(Ф(ψ(Ф(2)Ф(2)))) = ψ(I(0,ψ(I(0,1)I(0,1))))
{n,n(1,,1(1(1(1(1''2'')2')2)2)2)2}
{n,n(1(2/1(1(1(1/33)2)2)22)2)2}
536FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(εΩ2+1)) = ψ(Ωψ(Ω3))
ψ(Ф(ψ(Ф(3)))) = ψ(I(0,ψ(I(0,2))))
{n,n(1,,1(1(1(1(1'''2'')2')2)2)2)2} = {n,n(1,,1(1(1,,4)2)2)2}
{n,n(1(2/1(1(1(1(1/32/42)2)2)2)22)2)2}
537FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(ГΩ2+1)) = ψ(Ωψ(Ω3Ω3))
ψ(Ф(ψ(Ф(3)Ф(3)))) = ψ(I(0,ψ(I(0,2)I(0,2))))
{n,n(1,,1(1(1(1(1(1''2''')2'')2')2)2)2)2}
{n,n(1(2/1(1(1(1(1/43)2)2)2)22)2)2}
538FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(Ωω))
ψ(Ф(ψ(Ф(ω)))) = ψ(I(0,ψ(I(0,ω))))
{n,n(1,,1(1(1,,1,2)2)2)2}
{n,n(1(2/1(1(2/1,22)2)22)2)2}
539FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(Ωψ(Ωω)))
ψ(Ф(ψ(Ф(ψ(Ф(ω)))))) = ψ(I(0,ψ(I(0,ψ(I(0,ω))))))
{n,n(1,,1(1(1,,1(1(1,,1,2)2)2)2)2)2}
{n,n(1(2/1(1(2/1(1(2/1,22)2)22)2)22)2)2}
540FGH ψ-function
FGH ψ(higher)
SAN
BAN
ψ(Ωψ(Ωψ(Ωψ(Ωω))))
ψ(Ф(ψ(Ф(ψ(Ф(ψ(Ф(ω)))))))) = ψ(I(0,ψ(I(0,ψ(I(0,ψ(I(0,ω))))))))
{n,n(1,,1(1(1,,1(1(1,,1(1(1,,1,2)2)2)2)2)2)2)2}
{n,n(1(2/1(1(2/1(1(2/1(1(2/1,22)2)22)2)22)2)22)2)2}
541FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
BAN
ψ(ΩΩ) = ψ(Ωψ(Ωψ(Ωψ(Ωψ(Ω...)))))
ψ(ω1-th uncountable)
ψ(Ф(Ф(1))) = ψ(Ф(ψ(Ф(ψ(Ф(ψ(Ф(...)))))))) = ψ(I(0,I(0,0))) = ψ(I(0,ψ(I(0,ψ(I(0,ψ(I(0,...))))))))
{n,n(1,,1'2)2} = {n,n(1,,1(1,,2)2)2}
{n,n(1(2/1(1(2/1(1(2/1(1(2/...2)2)22)2)22)2)22)2)2}
542FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩω)
ψ(Ф(Ф(1))+Ф(ω)) = ψ(I(0,I(0,0))+I(0,ω))
{n,n(1(1,,1,2)2,,1'2)2}
543FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩω+1) = ψ(ΩΩΩω+1)
ψ(Ф(Ф(1))+Ф(ω+1)) = ψ(I(0,I(0,0))+I(0,ω+1))
{n,n(1(1,,2,2)2,,1'2)2}
544FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ×2)
ψ(Ф(Ф(1))+Ф(Ф(1))) = ψ(I(0,I(0,0))+I(0,I(0,1)))
{n,n(1(1,,1'2)2,,1'2)2}
545FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ2)
ψ(Ф(Ф(1))2) = ψ(I(0,I(0,0))2) = ψ(χ(χ(0))2)
{n,n(1(1,,1'2)1(1,,1'2)2,,1'2)2}
546FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩΩΩ)
ψ(Ф(Ф(1))Ф(Ф(1))) = ψ(I(0,I(0,0))I(0,I(0,0)))
{n,n(1(1(1,,1'2)2,,1'2)2,,1'2)2}
547FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩΩΩΩΩ)
ψ(Ф(Ф(1))Ф(Ф(1))Ф(Ф(1))) = ψ(I(0,I(0,0))I(0,I(0,0))I(0,I(0,0)))
{n,n(1(1(1,,1'2)1(1,,1'2)2,,1'2)2,,1'2)2}
548FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ+1) = ψ(εΩΩ+1) = ψ(ΩΩΩΩΩΩ...)
ψ(Ф(Ф(1)+1)) = ψ(I(0,I(0,0)+1))
{n,n(1,,2'2)2}
549FGH ψ-function
FGH ψ(higher)
SAN
ψ(ГΩΩ+1) = ψ(ΩΩ+1ΩΩ+1)
ψ(Ф(Ф(1)+1)Ф(Ф(1)+1)) = ψ(I(0,I(0,0)+1)I(0,I(0,0)+1))
{n,n(1(1(1,,2'2)2,,2'2)2,,2'2)2}
550FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ+ψ(ΩΩ))
ψ(Ф(Ф(1)+ψ(Ф(Ф(1))))) = ψ(I(0,I(0,0)+ψ(I(0,I(0,0)))))
{n,n(1,,1(1(1,,1'2)2)2'2)2}
551FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ)))
ψ(Ф(Ф(1)+ψ(Ф(Ф(1)+ψ(Ф(Ф(1))))))) = ψ(I(0,I(0,0)+ψ(I(0,I(0,0)+ψ(I(0,I(0,0)))))))
{n,n(1,,1(1(1,,1(1(1,,1'2)2)2'2)2)2'2)2}
552FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩ×2) = ψ(ΩΩ+ψ(ΩΩ+ψ(ΩΩ+...)))
ψ(Ф(Ф(1)+Ф(1))) = ψ(I(0,I(0,0)+I(0,0)))
{n,n(1,,1'3)2}
553FGH ψ-function
FGH ψ(higher)
SAN
ψ(Ωψ12)) = ψ(ΩεΩ+1)
ψ(Ф(ψФ(2)(Ф(2)))) = ψ(I(0,(ψI(0,1)(I(0,0))))
{n,n(1,,1(1''2,,2)2)2}
554FGH ψ-function
FGH ψ(higher)
SAN
ψ(Ωψ1Ω))
ψ(Ф(ψФ(2)(Ф(Ф(1))))) = ψ(I(0,(ψI(0,1)(I(0,I(0,0)))))
{n,n(1,,1(1(1,,1'2)2,,2)2)2}
555FGH ψ-function
FGH ψ(higher)
SAN
ψ(Ωψ1ψ12))) = ψ(ΩΩεΩ+1)
ψ(Ф(ψФ(2)(Ф(ψФ(2)(Ф(2)))))) = ψ(I(0,(ψI(0,1)(I(0,ψI(0,1)(I(0,0))))))
{n,n(1,,1(1(1,,1(1''2,,2)2)2,,2)2)2}
556FGH ψ-function
FGH ψ(higher)
SAN
ψ(Ωψ1ψ1Ω)))
ψ(Ф(ψФ(2)(Ф(ψФ(2)(Ф(Ф(1))))))) = ψ(I(0,(ψI(0,1)(I(0,ψI(0,1)(I(0,I(0,0)))))))
{n,n(1,,1(1(1,,1(1(1,,1'2)2,,2)2)2,,2)2)2}
557FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩ2) = ψ(Ωψ1ψ1ψ1(...))))
ψ(ω2-th uncountable)
ψ(Ф(Ф(2))) = ψ(I(0,I(0,1)))
{n,n(1,,1''2)2} = {n,n(1,,1(1,,3)2)2}
558FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩ3)
ψ(ω3-th uncountable)
ψ(Ф(Ф(3))) = ψ(I(0,I(0,2)))
{n,n(1,,1'''2)2} = {n,n(1,,1(1,,4)2)2}
559FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩω)
ψ(ωω-th uncountable)
ψ(Ф(Ф(ω))) = ψ(I(0,I(0,ω)))
{n,n(1,,1(1,,1,2)2)2}
560FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩω+1) = ψ(ΩεΩω+1)
ψ(Ф(Ф(ω+1))) = ψ(I(0,I(0,ω+1)))
{n,n(1,,1(1,,2,2)2)2}
561FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩω×2)
ψ(Ф(Ф(ω×2))) = ψ(I(0,I(0,ω×2)))
{n,n(1,,1(1,,1,3)2)2}
562FGH ψ-function
FGH ψ(higher)
SAN
ψ(ΩΩε0) = ψ(ΩΩψ(Ω))
ψ(Ф(Ф(ψ(Ф(1)))))) = ψ(I(0,I(0,ψ(I(0,0)))))
{n,n(1,,1(1,,1(1'2)2)2)2}
563FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩΩ) = ψ(I(0,I(0,I(0,0))))
ψ(ωω1-th uncountable)
ψ(Ф(Ф(Ф(1)))) = ψ(I(0,I(0,I(0,0))))
{n,n(1,,1(1,,1'2)2)2}
564FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩΩω) = ψ(I(0,I(0,I(0,ω))))
ψ(ωωω-th uncountable)
ψ(Ф(Ф(Ф(ω)))) = ψ(I(0,I(0,I(0,ω))))
{n,n(1,,1(1,,1(1,,1,2)2)2)2}
565FGH ψ-function
ψ(cardinal)

FGH ψ(higher)
SAN
ψ(ΩΩΩΩω) = ψ(I(0,I(0,I(0,I(0,ω)))))
ψ(ωωωω-th uncountable)
ψ(Ф(Ф(Ф(Ф(Ф(ω)))))) = ψ(I(0,I(0,I(0,I(0,ω)))))
{n,n(1,,1(1,,1(1,,1(1,,1,2)2)2)2)2}
566proof theory
FGH ψ-function

ψ(cardinal)
FGH ψ(ψM)

ψ(ψM(cardinal))
FGH Veblen Ф

FGH ψ(ψK)
SAN
П11-TR0
ψ(ΩΩΩΩ...) = ψΩαI(0)) = ψ(ψI(0)) = ψ(I(0,I(0,I(0,I(0,I(0,...)))))) = ψΩαI(1,0)(0))
ψ(1st fixed point of uncountable)
ψ(ψI(M)) = ψ(ψχ(M)(M)) = ψ(M) = ψ(χ(χ(χ(χ(...χ(sup(n))...)))))
ψ(1st weakly Mahlo)
ψ(Ф(1,0)) = ψ(Ф(Ф(Ф(Ф(Ф(...))))))
ψ(χK(1,0)) = ψ(χK(1,1)) = ψ(χK(1))
{n,n(1,,1,,2)2} =  {n,n(1,,1(1,,1,,2)2)2} ={n,n(1,,1...(1,,1,2)...2)2} ~n-1 insert
567FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)

SAN
ψ(Ф(1,0)+1)
ψ(ψI(0)+1) = ψ(ψI(0))×ω
ψ(M+1)
{n,n(1,2,,1(1,,1,,2)2)2}
568FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(1))
ψ(ψI(0)+Ω) = ψ(ψI(0)+ψ(ψI(0)+ψ(ψI(0)+...))) = εψ(ψI(0))+1 = ψ(ψI(0))ψ(ψI(0))ψ(ψI(0))...
ψ(M+Ω)
{n,n(1'2,,1(1,,1,,2)2)2}
569FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(2))
ψ(ψI(0)+Ω2) = ψ(ψI(0)+ψ1I(0)+ψ1I(0)+...)))
ψ(M+Ω2)
{n,n(1(1,,3)2,,1(1,,1,,2)2)2}
570FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(ω))
ψ(ψI(0)+Ωω)
ψ(M+Ωω)
{n,n(1(1,,1,2)2,,1(1,,1,,2)2)2}
571FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(Ф(1)))
ψ(ψI(0)+ΩΩ)
ψ(M+ΩΩ)
{n,n(1(1,,1'2)2,,1(1,,1,,2)2)2}
572FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(Ф(ω)))
ψ(ψI(0)+ΩΩω)
ψ(M+ΩΩω)
{n,n(1(1,,1(1,,1,2)2)2,,1(1,,1,,2)2)2}
573FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)+Ф(Ф(Ф(ω))))
ψ(ψI(0)+ΩΩΩω)
ψ(M+ΩΩΩω)
{n,n(1(1,,1(1,,1(1,,1,2)2)2)2,,1(1,,1,,2)2)2}
574FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)×2) = ψ(Ф(1,0)+Ф(1,0)) = ψ(Ф(1,0)+Ф(Ф(Ф(Ф(...)))))
ψ(ψI(0)×2) = ψ(ψI(0)+ψI(0)) = ψ(ψI(0)+ΩΩΩΩ...)
ψ(M+ψχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2}
575FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)×3) = ψ(Ф(1,0)+Ф(1,0)+Ф(1,0))
ψ(ψI(0)×3) = ψ(ψI(0)+ψI(0)+ψI(0))
ψ(M+ψχ(M)(M)×2)
{n,n(1(1,,1(1,,1,,2)2)3,,1(1,,1,,2)2)2}
576FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)×ω)
ψ(ψI(0)×ω)
ψ(M+ψχ(M)(M)×ω)
{n,n(1(1,,1(1,,1,,2)2)1,2,,1(1,,1,,2)2)2}
577FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)×Ф(1))
ψ(ψI(0)×Ω) = ψ(ψI(0)×ψ(ψI(0)×ψ(ψI(0)×....)))
ψ(M+ψχ(M)(M)×Ω)
{n,n(1(1,,1(1,,1,,2)2)1'2,,1(1,,1,,2)2)2}
578FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)2)
ψ(ψI(0)2)
ψ(M+ψχ(M)(M)2)
{n,n(1(1,,1(1,,1,,2)2)1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2}
579FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)ω)
ψ(ψI(0)ω)
ψ(M+ψχ(M)(M)ω)
{n,n(1(2,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2}
580FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)Ф(1))
ψ(ψI(0)Ω) = ψ(ψI(0)ψ(ψI(0))ψ(ψI(0))...)
ψ(M+ψχ(M)(M)Ω)
{n,n(1(1'2,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2}
581FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0)Ф(1,0))
ψ(ψI(0)ψI(0))
ψ(M+ψχ(M)(M)ψχ(M)(M))
{n,n(1(1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2}
582FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)) = ψ(Ф(1,0)Ф(1,0)Ф(1,0)...)
ψ(ψI(0)ψI(0)ψI(0)...) = ψ(εψI(0)+1) = ψ(ΩψI(0)+1) = ψ(ΩΩΩΩ...+1)
ψ(1st uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)+1)
{n,n(1(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2} = {n,n(1,,2(1,,1,,2)2)2}
583FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)+Ф(ω))
ψ(ΩψI(0)+1ω)
ψ(M+Ωψχ(M)(M)+1ω)
{n,n(1(1,,1,2)2(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2}
584FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)+Ф(1,0))
ψ(ΩψI(0)+1I(0)) = ψ(ΩψI(0)+1ΩΩΩ...)
ψ(M+Ωψχ(M)(M)+1χ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2}
585FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)+Ф(1,0)Ф(1,0)Ф(1,0)...) = ψ(Ф(Ф(1,0)+1)+ψФ(Ф(1,0)+1)Ф(Ф(1,0)+1))
ψ(ΩψI(0)+1ΩψI(0)+1ψI(0)+1)) = ψ(ΩψI(0)+1I(0)ψI(0)ψI(0)...)
ψ(M+Ωψχ(M)(M)+1Ωψχ(M)(M)+1ψχ(M)(M)+1))
{n,n(1(1(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2}
586FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)×2) = ψ(Ф(Ф(1,0)+1)+Ф(Ф(1,0)+1)) = ψ(Ф(Ф(1,0)+1)+ψФ(Ф(1,0)+1)(Ф(Ф(1,0)+ψФ(Ф(1,0)+1)(...)))))
ψ(ΩψI(0)+1×2) = ψ(ΩψI(0)+1ψI(0)+1) = ψ(ΩψI(0)+1ΩψI(0)+1ψI(0)+1ΩψI(0)+1(...)))
ψ(M+Ωψχ(M)(M)+1×2)
{n,n(1(1,,2(1,,1,,2)2)3,,1(1,,1,,2)2)2}
587FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)2)) = ψ(Ф(Ф(1,0)+1)×Ф(Ф(1,0)+1)) = ψ(Ф(Ф(1,0)+1)×ψФ(Ф(1,0)+1)(Ф(Ф(1,0)×ψФ(Ф(1,0)+1)(...)))))
ψ(ΩψI(0)+12) = ψ(ΩψI(0)+1×ΩψI(0)+1) = ψ(ΩψI(0)+1×ψΩψI(0)+1ψI(0)+1×ψΩψI(0)+1(...)))
ψ(M+Ωψχ(M)(M)+12)
{n,n(1(1,,2(1,,1,,2)2)1(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2}
588FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+1)Ф(Ф(1,0)+1)))) = ψ(Ф(Ф(1,0)+1)ψФ(Ф(1,0)+1)(Ф(Ф(1,0)+1)ψФ(Ф(1,0)+1)(...)))))
ψ(ΩψI(0)+1ΩψI(0)+1) = ψ(ΩψI(0)+1ψΩψI(0)+1ψI(0)+1ψΩψI(0)+1(...)))
ψ(M+Ωψχ(M)(M)+1Ωψχ(M)(M)+1)
{n,n(1(1(1,,2(1,,1,,2)2)2,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2}
589FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+2)) = ψ(Ф(Ф(1,0)+1)Ф(Ф(1,0)+1)...))
ψ(ΩψI(0)+2) = ψ(ΩψI(0)+1ΩψI(0)+1...) = ψ(εΩψI(0)+1)
ψ(2nd uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)+2)
{n,n(1(1(1,,3(1,,1,,2)2)2,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2} = {n,n(1,,3(1,,1,,2)2)2}
590FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+ω))
ψ(ΩψI(0)+ω)
ψ(ω-th uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)+ω)
{n,n(1,,1,2(1,,1,,2)2)2}
591FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+Ф(1)))
ψ(ΩψI(0)+Ω) = ψ(ΩψI(0)+ψ(ΩψI(0)+...))
ψ(ω1-th uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)+Ω)
{n,n(1,,1'2(1,,1,,2)2)2}
592FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)+Ф(ω)))
ψ(ΩψI(0)+Ωω)
ψ(ωω-th uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)+Ωω)
{n,n(1,,1(1,,1,2)2(1,,1,,2)2)2}
593FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)×2)) = ψ(Ф(Ф(1,0)+Ф(1,0)))
ψ(ΩψI(0)×2)
ψ(1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+Ωψχ(M)(M)×2)
{n,n(1,,1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2}
594FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)×2+1)) = ψ(Ф(Ф(1,0)×2)Ф(Ф(1,0)×2)...))
ψ(ΩψI(0)×2+1) = ψ(ΩψI(0)×2ΩψI(0)×2...) = ψ(εΩψI(0)×2)
ψ(M+Ωψχ(M)(M)×2+1)
{n,n(1,,2(1,,1(1,,1,,2)2)2(1,,1,,2)2)2}
595FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)×3))
ψ(ΩψI(0)×3)
ψ(M+Ωψχ(M)(M)×3)
{n,n(1,,1(1,,1(1,,1,,2)2)3(1,,1,,2)2)2}
596FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)×ω))
ψ(ΩψI(0)×ω)
ψ(M+Ωψχ(M)(M)×ω)
{n,n(1,,1(1,,1(1,,1,,2)2)1,2(1,,1,,2)2)2}
597FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)×Ф(1)))
ψ(ΩψI(0)×Ω) = ψ(ΩψI(0)×ψ(ΩψI(0)×...))
ψ(M+Ωψχ(M)(M)×Ω)
{n,n(1,,1(1,,1(1,,1,,2)2)1'2(1,,1,,2)2)2}
598FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)2)) = ψ(Ф(Ф(1,0)×Ф(1,0)))
ψ(ΩψI(0)2)
ψ(M+Ωψχ(M)(M)2)
{n,n(1,,1(1,,1(1,,1,,2)2)1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2}
599FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)ω))
ψ(ΩψI(0)ω)
ψ(M+Ωψχ(M)(M)ω)
{n,n(1,,1(2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
600FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)Ф(1)))
ψ(ΩψI(0)Ω) = ψ(ΩψI(0)ψ(ΩψI(0)...))
ψ(M+Ωψχ(M)(M)Ω)
{n,n(1,,1(1'2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
601FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0)Ф(1,0)))
ψ(ΩψI(0)ψI(0))
ψ(M+Ωψχ(M)(M)ψχ(M)(M))
{n,n(1,,1(1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
602FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)+1)))) = ψ(Ф(Ф(1,0)Ф(1,0)Ф(1,0)...))
ψ(ΩψΩψI(0)+1ψI(0)+1)) = ψ(ΩεψI(0)+1) = ψ(ΩψI(0)ψI(0)ψI(0)...)
ψ(M+ΩψΩψχ(M)(M)+1ψχ(M)(M)+1))
{n,n(1,,1(1(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
603FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)+2))))
ψ(ΩψΩψI(0)+1ψI(0)+2))
ψ(M+ΩψΩψχ(M)(M)+1ψχ(M)(M)+2))
{n,n(1,,1(1(1,,3(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
604FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)×2))))
ψ(ΩψΩψI(0)+1ψI(0)×2))
ψ(M+ΩψΩψχ(M)(M)+1ψχ(M)(M)×2))
{n,n(1,,1(1(1,,1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
605FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)2))))
ψ(ΩψΩψI(0)+1ψI(0)2))
ψ(M+ΩψΩψχ(M)(M)+1ψχ(M)(M)2))
{n,n(1,,1(1(1,,1(1,,1(1,,1,,2)2)1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
606FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)Ф(1,0)))))
ψ(ΩψΩψI(0)+1ψI(0)ψI(0)))
ψ(M+ΩψΩψχ(M)(M)+1ψχ(M)(M)ψχ(M)(M)))
{n,n(1,,1(1(1,,1(1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
607FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+1)(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)+1)))))) = ψ(Ф(ψФ(Ф(1,0)+1)(Ф(Ф(1,0)Ф(1,0)Ф(1,0)...))))
ψ(ΩψΩψI(0)+1(ΩψΩψI(0)+1ψI(0)+1))) = ψ(ΩψΩψI(0)+1εψI(0)+1)) = ψ(ΩψΩψI(0)+1ψI(0)ψI(0)ψI(0)...))
ψ(M+ΩψΩψχ(M)(M)+1(ΩψΩψχ(M)(M)+1ψχ(M)(M)+1)))
{n,n(1,,1(1(1,,1(1(1,,2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2,,1(1,,1,,2)2)2(1,,1,,2)2)2}
608FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+1))) = ψ(Ф(ψФ(Ф(1,0)+1)(Ф(ψФ(Ф(1,0)+1)(Ф(ψФ(Ф(1,0)+1)(...)))))))
ψ(ΩΩψI(0)+1) = ψ(ΩΩΩΩ...+1) = ψ(ΩψΩψI(0)+1ψΩψI(0)+1ψΩψI(0)+1(...))))
ψ(1st uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)+1)
{n,n(1,,1(1,,2(1,,1,,2)2)2(1,,1,,2)2)2}
609FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+1)+1))
ψ(ΩΩψI(0)+1+1) = ψ(ΩΩΩΩ...+1+1)
ψ(M+ΩΩψχ(M)(M)+1+1)
{n,n(1,,2(1,,2(1,,1,,2)2)2(1,,1,,2)2)2}
610FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+1)+ψФ(Ф(1,0)+1)Ф(Ф(Ф(1,0)+1))))
ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1))
ψ(M+ΩΩψχ(M)(M)+1Ωψχ(M)(M)+1Ωψχ(M)(M)+1))
{n,n(1,,1(1,,1(1,,2(1,,1,,2)2)2(1,,1,,2)2)2(1,,2(1,,1,,2)2)2(1,,1,,2)2)2}
611FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+1)×2)) = ψ(Ф(Ф(Ф(1,0)+1)+ψФ(Ф(1,0)+1)Ф(Ф(Ф(1,0)+1)+...)))
ψ(ΩΩψI(0)+1×2) = ψ(ΩΩψI(0)+1ψI(0)+1) = ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1+...))
ψ(M+ΩΩψχ(M)(M)+1×2)
{n,n(1,,1(1,,2(1,,1,,2)2)3(1,,1,,2)2)2}
612FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ψФ(Ф(1,0)+2)(Ф(Ф(1,0)+2)))) = ψ(Ф(Ф(Ф(1,0)+1)Ф(Ф(1,0)+1)Ф(Ф(1,0)+1)...))
ψ(ΩψΩψI(0)+2ψI(0)+2)) = ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1...)
ψ(M+ΩψΩψχ(M)(M)+2ψχ(M)(M)+2))
{n,n(1,,1(1(1,,3(1,,1,,2)2)2,,2(1,,1,,2)2)2(1,,1,,2)2)2}
613FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+2))) = ψ(Ф(ψФ(Ф(1,0)+2)(Ф(ψФ(Ф(1,0)+2)(Ф(Ф(ψФ(Ф(1,0)+2)(...))))))))
ψ(ΩΩψI(0)+2) = ψ(ΩψΩψI(0)+2ψΩψI(0)+2ψΩψI(0)+2(...))))
ψ(2nd uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)+2)
{n,n(1,,1(1,,3(1,,1,,2)2)2(1,,1,,2)2)2}
614FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+ω)))
ψ(ΩΩψI(0)+ω)
ψ(ω-th uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)+ω)
{n,n(1,,1(1,,1,2(1,,1,,2)2)2(1,,1,,2)2)2}
615FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+Ф(1))))
ψ(ΩΩψI(0)+Ω) = ψ(ΩΩψI(0)+ψ(ΩΩψI(0)+...))
ψ(ω1-th uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)+Ω)
{n,n(1,,1(1,,1'2(1,,1,,2)2)2(1,,1,,2)2)2}
616FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)+Ф(ω))))
ψ(ΩΩψI(0)+Ωω)
ψ(ωω-th uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)+Ωω)
{n,n(1,,1(1,,1(1,,1,2)2(1,,1,,2)2)2(1,,1,,2)2)2}
617FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0)×2))) = ψ(Ф(Ф(Ф(1,0)+Ф(1,0))))
ψ(ΩΩψI(0)×2) = ψ(ΩΩψI(0)+ψI(0))
ψ(1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩψχ(M)(M)×2)
{n,n(1,,1(1,,1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2}
618FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(Ф(1,0)+1))))
ψ(ΩΩΩψI(0)+1)
ψ(1st uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable-th uncountable after 1st fixed point of uncountable)
ψ(M+ΩΩΩψχ(M)(M)+1)
{n,n(1,,1(1,,1(1,,2(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2}
619FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(Ф(1,0)×2))))
ψ(ΩΩΩψI(0)×2)
ψ(M+ΩΩΩψχ(M)(M)×2)
{n,n(1,,1(1,,1(1,,1(1,,1(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2}
620FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(Ф(Ф(1,0)+1)))))
ψ(ΩΩΩΩψI(0)+1)
ψ(M+ΩΩΩΩψχ(M)(M)+1)
{n,n(1,,1(1,,1(1,,1(1,,2(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2(1,,1,,2)2)2}
621FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(1,1)) = ψ(Ф(Ф(Ф(Ф(...Ф(1,0)+1...)))))
ψ(ψI(1)) = ψ(ΩΩΩ...ψI(0)+1)
ψ(2nd fixed point of uncountable)
ψ(M+ψχ(M)(M+1)) = ψ(M+ΩΩΩ...ψχ(M)(M)+1)
{n,n(1,,1(1,,1,,2)3)2}
622FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,1)+1))
ψ(ΩψI(1)+1)
ψ(M+Ωψχ(M)(M+1)+1)
{n,n(1,,2(1,,1,,2)3)2}
623FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,1)+Ф(1,0)))
ψ(ΩψI(1)+ψI(0))
ψ(M+Ωψχ(M)(M+1)+ψχ(M)(M))
{n,n(1,,1(1,,1(1,,1,,2)2)2(1,,1,,2)3)2}
624FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,1)×2))
ψ(ΩψI(1)×2)
ψ(M+Ωψχ(M)(M+1)×2)
{n,n(1,,1(1,,1(1,,1,,2)3)2(1,,1,,2)3)2}
625FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,1)+1)))
ψ(ΩΩψI(1)+1)
ψ(M+ΩΩψχ(M)(M+1)+1)
{n,n(1,,1(1,,2(1,,1,,2)3)2(1,,1,,2)3)2}
626FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,1)×2)))
ψ(ΩΩψI(1)×2)
ψ(M+ΩΩψχ(M)(M+1)×2)
{n,n(1,,1(1,,1(1,,1(1,,1,,2)3)2(1,,1,,2)3)2(1,,1,,2)3)2}
627FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(1,1)) = ψ(Ф(Ф(Ф(Ф(...Ф(1,1)+1...)))))
ψ(ψI(2)) = ψ(ΩΩΩ...ψI(1)+1)
ψ(3d fixed point of uncountable)
ψ(M+ψχ(M)(M+2)) = ψ(M+ΩΩΩ...ψχ(M)(M+1)+1)
{n,n(1,,1(1,,1,,2)4)2}
628FGH Veblen Ф
FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Ф(1,ω))
ψ(ψI(ω))
ψ(ω-th fixed point of uncountable)
ψ(M+ψχ(M)(M+ω))
{n,n(1,,1(1,,1,,2)1,2)2}
629FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1)))
ψ(ψI(Ω)) = ψ(ψI(ψ(ψI(ψ(ψI(ψ(ψI(...))))))))
ψ(M+ψχ(M)(M+Ω))
{n,n(1,,1(1,,1,,2)1'2)2}
630FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(2)))
ψ(ψI2)) = ψ(ψI1I1I1I(...))))))))
ψ(M+ψχ(M)(M+Ω2))
{n,n(1,,1(1,,1,,2)1(1,,3)2)2}
631FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(ω)))
ψ(ψIω)) = ψ(ψIωIωIωI(...))))))))
ψ(M+ψχ(M)(M+Ωω))
{n,n(1,,1(1,,1,,2)1(1,,1,2)2)2}
632FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0)))
ψ(ψII(0))) = ψ(ψIΩΩΩ...))
ψ(M+ψχ(M)(M+ψχ(M)(M)))
{n,n(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2)2)2}
633FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,Ф(1,0))+1))
ψ(ΩψII(0))+1)
ψ(M+Ωψχ(M)(M+ψχ(M)(M))+1)
{n,n(1,,2(1,,1,,2)1(1,,1(1,,1,,2)2)2)2}
634FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,Ф(1,0))×2))
ψ(ΩψII(0))×2)
ψ(M+Ωψχ(M)(M+ψχ(M)(M))×2)
{n,n(1,,1(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2)2)2(1,,1,,2)1(1,,1(1,,1,,2)2)2)2}
635FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0)+1))
ψ(ψII(0)+1))
ψ(M+ψχ(M)(M+ψχ(M)(M)+1))
{n,n(1,,1(1,,1,,2)2(1,,1(1,,1,,2)2)2)2}
636FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0)×2))
ψ(ψII(0)×2))
ψ(M+ψχ(M)(M+ψχ(M)(M)×2))
{n,n(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2)3)2}
637FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(Ф(1,0)+1)))
ψ(ψIψI(0)+1))
ψ(M+ψχ(M)(M+Ωψχ(M)(M)+1))
{n,n(1,,1(1,,1,,2)1(1,,2(1,,1,,2)2)2)2}
638FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,1)))
ψ(ψII(1)))
ψ(M+ψχ(M)(M+ψχ(M)(M+1)))
{n,n(1,,1(1,,1,,2)1(1,,1(1,,1,,2)3)2)2}
639FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,Ф(1,0))))
ψ(ψIII(0))))
ψ(M+ψχ(M)(M+ψχ(M)(M+ψχ(M)(M))))
{n,n(1,,1(1,,1,,2)1(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2)2)2)2}
640FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
ψ(cardinal)

SAN
ψ(Ф(2,0)) = ψ(Ф(1,Ф(1,Ф(1,Ф(1,Ф(1,...))))))
ψ(I) = ψΩαI(I)) = ψ(ψI(I)) = ψ(ψIIII(...))))) = ψ(I(1,0))
ψ(M+χ(M)) = ψ(M+ψχ(M)(M+ψχ(M)(M+ψχ(M)(M+...))))
ψ(1st weakly inaccessible)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2)2}
641FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+1) = ψ(Ф(2,0))×ω = ψ(Ф(1,Ф(1,Ф(1,Ф(1,Ф(1,...)))))+1)
ψ(I+1) = ψ(I)×ω = ψ(ψIIII(...))))+1)
ψ(M+χ(M)+1)
{n,n(2,,1(1,,1,,2)1(1,,1,,2)2)2}
642FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(2,0)+1))
ψ(Ф(2,0)+Ф(1)) = ψ(Ф(2,0))ψ(Ф(2,0))ψ(Ф(2,0))... = ψ(I)ψ(I)ψ(I)...
ψ(I+Ω) = ψ(ψI(I+ψ(ψI(I+ψ(ψI(I+ψ(ψI(I+...))))))))
ψ(M+χ(M)+Ω) = ψ(M+χ(M)+ψ(M+χ(M)+ψ(M+χ(M)+...)))
{n,n(1'2,,1(1,,1,,2)1(1,,1,,2)2)2}
643FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(2))
ψ(I+Ω2) = ψ(ψI(I+ψ2I(I+ψ2I(I+ψ2I(I+...))))))))
ψ(M+χ(M)+Ω2)
{n,n(1(1,,3)2,,1(1,,1,,2)1(1,,1,,2)2)2}
644FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(ω))
ψ(I+Ωω) = ψ(ψI(I+ψωI(I+ψωI(I+ψωI(I+...))))))))
ψ(M+χ(M)+Ωω)
{n,n(1(1,,1,2)2,,1(1,,1,,2)1(1,,1,,2)2)2}
645FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(1,0))
ψ(I+ψI(0))
ψ(M+χ(M)+ψχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1,,1,,2)1(1,,1,,2)2)2}
646FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(1,1))
ψ(I+ψI(1))
ψ(M+χ(M)+ψχ(M)(M+1))
{n,n(1(1,,1(1,,1,,2)3)2,,1(1,,1,,2)1(1,,1,,2)2)2}
647FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(1,Ф(1,0)))
ψ(I+ψII(0)))
ψ(M+χ(M)+ψχ(M)(M+ψχ(M)(M)))
{n,n(1(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2)2)2,,1(1,,1,,2)1(1,,1,,2)2)2}
648FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(2,0))
ψ(I+ψI(I)) = ψ(I+ψIIII(...)))))
ψ(M+χ(M)+ψχ(M)(M+χ(M)))
{n,n(1(1,,1(1,,1,,2)1(1,,1,,2)2)2,,1(1,,1,,2)1(1,,1,,2)2)2}
649FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)+Ф(2,0)+Ф(2,0))
ψ(I+ψI(I)×2)
ψ(M+χ(M)+ψχ(M)(M+χ(M))×2)
{n,n(1(1,,1(1,,1,,2)1(1,,1,,2)2)3,,1(1,,1,,2)1(1,,1,,2)2)2}
650FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0)×ω)
ψ(I+ψI(I)×ω)
ψ(M+χ(M)+ψχ(M)(M+χ(M))×ω)
{n,n(1(1,,1(1,,1,,2)1(1,,1,,2)2)1,2,,1(1,,1,,2)1(1,,1,,2)2)2}
651FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(2,0)+1)) = ψ(Ф(2,0)Ф(2,0)Ф(2,0)...)
ψ(I+ψI(I)ψI(I)ψI(I)...) = ψ(I+εψI(I)+1) = ψ(I+ΩψI(I)+1)
ψ(M+χ(M)+Ωψχ(M)(M+χ(M))+1)
{n,n(1,,2(1,,1,,2)1(1,,1,,2)2)2}
652FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(2,0)+2))
ψ(I+ΩψI(I)+2)
ψ(M+χ(M)+Ωψχ(M)(M+χ(M))+2)
{n,n(1,,3(1,,1,,2)1(1,,1,,2)2)2}
653FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(2,0)+Ф(2,0)))
ψ(I+ΩψI(I)×2)
ψ(M+χ(M)+Ωψχ(M)(M+χ(M))×2)
{n,n(1,,1(1,,1(1,,1,,2)1(1,,1,,2)2)2(1,,1,,2)1(1,,1,,2)2)2}
654FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(...Ф(2,0)+1...))))) = ψ(Ф(1,Ф(2,0)+1))
ψ(I+ψI(I+1)) = ψ(I+ΩΩΩ...ψI(I)+1) = ψ(ψIIII(...)))+1))
ψ(M+χ(M)+ψχ(M)(M+χ(M)+1))
{n,n(1,,1(1,,1,,2)2(1,,1,,2)2)2}
655FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(Ф(Ф(...Ф(2,0)+1...)))))) = ψ(Ф(1,Ф(1,Ф(2,0)+1))))
ψ(I+ψI(I+ψI(I+1))) = ψ(I+ψI(I+ΩΩΩ...ψI(I)+1)) = ψ(ψIIII(...))+1)))
ψ(M+χ(M)+ψχ(M)(M+χ(M)+ψχ(M)(M+χ(M)+1)))
{n,n(1,,1(1,,1,,2)1(1,,1(1,,1,,2)2(1,,1,,2)2)2(1,,1,,2)2)2}
656FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,1)) = ψ(Ф(1,Ф(1,Ф(1,Ф(1,...Ф(2,0)+1...)))))
ψ(I×2) = ψ(I+I) = ψ(I+ψI(I+ψI(I+ψI(I+...)))) = ψ(ψIIII(...ψI(I)+1...)))))
ψ(M+χ(M)+χ(M)) = ψ(M+χ(M)×2)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)3)2}
657FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,ω))
ψ(I×ω)
ψ(M+χ(M)×ω)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1,2)2}
658FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,Ф(1)))
ψ(I×Ω) = ψ(I×ψ(ψI(I×ψ(ψI(I×...)))))
ψ(M+χ(M)×Ω) = ψ(M+χ(M)×ψ(M+χ(M)×ψ(M+χ(M)×...)))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1'2)2}
659FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,Ф(1,0)))
ψ(I×ψI(0)) = ψ(I×ΩΩΩΩ...)
ψ(M+χ(M)×ψχ(M)(M))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1(1,,1(1,,1,,2)2)2)2}
660FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,Ф(2,0)))
ψ(I×ψI(I))
ψ(M+χ(M)×ψχ(M)(M+χ(M)))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1(1,,1(1,,1,,2)1(1,,1,,2)2)2)2}
661FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(3,0)) = ψ(Ф(2,Ф(2,Ф(2,Ф(2,Ф(2,...))))))
ψ(I2) = ψ(I×I) = ψ(I×ψI(I×ψI(I×ψI(I×...))))
ψ(M+χ(M)×χ(M)) = ψ(M+χ(M)2)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1(1,,1,,2)2)2}
662FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(4,0)) = ψ(Ф(3,Ф(3,Ф(3,Ф(3,Ф(3,...))))))
ψ(I3)
ψ(M+χ(M)3)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1(1,,1,,2)1(1,,1,,2)2)2}
663FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ω,0))
ψ(Iω)
ψ(M+χ(M)ω)
{n,n(1,,1(2,,1,,2)2)2} = {n,n(2,,1,,2)2}
664FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1),0))
ψ(IΩ) = ψ(Iψ(Iψ(I...)))
ψ(M+χ(M)Ω)
{n,n(1,,1(1'2,,1,,2)2)2}
665FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(1,0),0))
ψ(IψI(0))
ψ(M+χ(M)ψχ(M)(M))
{n,n(1,,1(1(1,,1(1,,1,,2)2)2,,1,,2)2)2}
666FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(2,0),0))
ψ(IψI(I))
ψ(M+χ(M)ψχ(M)(M+χ(M)))
{n,n(1,,1(1(1,,1(1,,1,,2)1(1,,1,,2)2)2,,1,,2)2)2}
667FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(ω,0),0))
ψ(IψI(Iω))
ψ(M+χ(M)ψχ(M)(M+χ(M)ω))
{n,n(1,,1(1(1,,1(2,,1,,2)2)2,,1,,2)2)2}
668FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0,0)) = ψ(Ф(Ф(Ф(Ф(...,0),0),0),0))
ψ(II) = ψ(IψI(IψI(I...)))
ψ(M+χ(M)χ(M))
{n,n(1,,1(1(1,,1,,2)2,,1,,2)2)2}
669FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0,0)+1))
ψ(II+1) = ψ(ΩΩΩ...ψI(II)+1)
ψ(M+χ(M)χ(M)+1)
{n,n(2,,1(1,,1,,2)1(1,,1,,2)2)2}
670FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0,0)+Ф(1)))
ψ(II+Ω) = ψ(II+ψ(II+ψ(II+...)))
ψ(M+χ(M)χ(M)+Ω)
{n,n(1'2,,1(1(1,,1,,2)2,,1,,2)2)2}
671FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,0,0)+Ф(1,0,0)))
ψ(III(II))
ψ(M+χ(M)χ(M)χ(M)(M+χ(M)χ(M)))
{n,n(1(1,,1(1(1,,1,,2)2,,1,,2)2)2,,1(1(1,,1,,2)2,,1,,2)2)2}
672FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(Ф(1,0,0)+1)))
ψ(IIψI(II)+1)
ψ(M+χ(M)χ(M)ψχ(M)(M+χ(M)χ(M))+1)
{n,n(1,,2(1(1,,1,,2)2,,1,,2)2)2}
673FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(Ф(1,0,0)+Ф(1,0,0))))
ψ(IIψI(II)×2)
ψ(M+χ(M)χ(M)ψχ(M)(M+χ(M)χ(M))×2)
{n,n(1,,1(1,,1(1(1,,1,,2)2,,1,,2)2)2(1(1,,1,,2)2,,1,,2)2)2}
674FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,Ф(1,0,0)+1))
ψ(III(II+1))
ψ(M+χ(M)χ(M)χ(M)(M+χ(M)χ(M)+1))
{n,n(1,,1(1,,1,,2)2(1(1,,1,,2)2,,1,,2)2)2}
675FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,Ф(1,Ф(1,0,0)+Ф(1,0,0)))
ψ(III(III(II)))
ψ(M+χ(M)χ(M)χ(M)(M+χ(M)χ(M)χ(M)(M+χ(M)χ(M)))))
{n,n(1,,1(1,,1,,2)1(1,,1(1(1,,1,,2)2,,1,,2)2)2(1(1,,1,,2)2,,1,,2)2)2}
676FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,Ф(1,0,0)+1))
ψ(II+I) = ψ(III(III(II+...)))
ψ(M+χ(M)χ(M)+χ(M))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1(1,,1,,2)2,,1,,2)2)2}
677FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,Ф(1,0,0)+2))
ψ(II+I×2)
ψ(M+χ(M)χ(M)+χ(M)×2)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)3(1(1,,1,,2)2,,1,,2)2)2}
678FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(3,Ф(1,0,0)+1))
ψ(II+I2)
ψ(M+χ(M)χ(M)+χ(M)2)
{n,n(1,,1(1,,1,,2)1(1,,1,,2)1(1,,1,,2)2(1(1,,1,,2)2,,1,,2)2)2}
679FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ω,Ф(1,0,0)+1))
ψ(II+Iω)
ψ(M+χ(M)χ(M)+χ(M)ω)
{n,n(1,,1(2,,1,,2)2(1(1,,1,,2)2,,1,,2)2)2}
680FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(Ф(Ф(1,0,0)+1)))
ψ(II+IψI(II+1))
ψ(M+χ(M)χ(M)+χ(M)ψχ(M)(M+χ(M)χ(M)+1))
{n,n(1,,1(1(2,,1(1(1,,1,,2)2,,1,,2)2)2,,1,,2)2(1(1,,1,,2)2,,1,,2)2)2}
681FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0,1)) = ψ(Ф(Ф(Ф(Ф(...Ф(1,0,0)+1...)))))
ψ(II×2) = ψ(II+II) = ψ(II+IψI(II+I...))
ψ(M+χ(M)χ(M)×2)
{n,n(1,,1(1(1,,1,,2)2,,1,,2)3)2}
682FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,1,0)) = ψ(Ф(1,0,Ф(1,0,Ф(1,0,Ф(1,0,...)))))
ψ(II+1)
ψ(M+χ(M)χ(M)+1)
{n,n(1,,1(1(1,,1,,2)2,,1,,2)1(1,,1,,2)2)2}
683FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(2,0,0)) = ψ(Ф(1,Ф(1,Ф(1,Ф(1,...,0),0),0),0))
ψ(II×2)
ψ(M+χ(M)χ(M)×2)
{n,n(1,,1(1(1,,1,,2)2,,1,,2)1(1(1,,1,,2)2,,1,,2)2)2}
684FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(ω,0,0))
ψ(II×ω)
ψ(M+χ(M)χ(M)×ω)
{n,n(1,,1(2(1,,1,,2)2,,1,,2)2)2}
685FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1,0,0,0)) = ψ(Ф(Ф(Ф(Ф(...,0,0),0,0),0,0),0,0))
ψ(II2)
ψ(M+χ(M)χ(M)2)
{n,n(1,,1(1(1,,1,,2)3,,1,,2)2)2}
686FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1ω)) = ψ(Ф(1,0,0,0,0,...))
ψ(IIω)
ψ(M+χ(M)χ(M)ω)
{n,n(1,,1(1(1,,1,,2)1,2,,1,,2)2)2}
687FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1Ф(1ω)))
ψ(IIψI(IIω))
ψ(M+χ(M)χ(M)ψχ(M)(M+χ(M)χ(M)ω))
{n,n(1,,1(1(1,,1,,2)1(1(1,,1,,2)1,2,,1,,2)2,,1,,2)2)2}
688FGH Veblen Ф
FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Ф(1Ф(1Ф(1Ф(1...)))))
ψ(III)
ψ(M+χ(M)χ(M)χ(M))
{n,n(1,,1(1(1,,1,,2)1(1,,1,,2)2,,1,,2)2)2}
689FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IIII)
ψ(M+χ(M)χ(M)χ(M)χ(M))
{n,n(1,,1(1(1(1,,1,,2)2,,1,,2)2,,1,,2)2)2}
690FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IIIII)
ψ(M+χ(M)χ(M)χ(M)χ(M)χ(M))
{n,n(1,,1(1(1(1,,1,,2)1(1,,1,,2)2,,1,,2)2,,1,,2)2)2}
691name
proof theory
FGH ψ-function

ψ(cardinal)
FGH ψ(ψM)
SAN
Small Rathjen ordinal
KPI
ψ(IIII...) = ψ(εI+1) = ψΩαIΩI+1I+1))) = ψ(ΩI+1) = ψ(I(0,I(1,0)+1))
ψ(1st uncountable after 1st weakly inaccessible)
ψ(M+χ(M+1)) = ψ(M+εχ(M)+1) = ψ(M+χ(M)χ(M)χ(M)χ(M)...)
{n,n(1,,1(1(1,,2,,2)2,,1,,2)2)2} = {n,n(1,,2,,2)2}
692FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1+1) = ψ(ΩI+1)×ω
ψ(M+χ(M+1)+1)
{n,n(2,,1(1(1,,2,,2)2,,1,,2)2)2}
693FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1+Ω) = ψ(ΩI+1+ψ(ΩI+1+ψ(ΩI+1+...))) = εψ(ΩI+1)+1 = ψ(ΩI+1)ψ(ΩI+1)ψI+1)...
ψ(M+χ(M+1)+Ω)
{n,n(1'2,,1(1(1,,2,,2)2,,1,,2)2)2}
694FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1I(0))
ψ(M+χ(M+1)+ψχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1(1,,2,,2)2,,1,,2)2)2}
695FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1I(I))
ψ(M+χ(M+1)+ψχ(M)(M+χ(M)))
{n,n(1(1,,1(1,,1,,2)1(1,,1,,2)2)2,,1(1(1,,2,,2)2,,1,,2)2)2}
696FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1II+1))
ψ(M+χ(M+1)+ψχ(M)(M+χ(M+1)))
{n,n(1(1,,1(1(1,,2,,2)2,,1,,2)2)2,,1(1(1,,2,,2)2,,1,,2)2)2}
697FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ψII+1)+1)
ψ(M+χ(M+1)+Ωψχ(M)(M+χ(M+1))+1)
{n,n(1,,2(1(1,,2,,2)2,,1,,2)2)2}
698FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1II+1+1)) = ψ(ΩI+1ΩΩ...ψII+1)+1)
ψ(M+χ(M+1)+ψχ(M)(M+χ(M+1)+1))
{n,n(1,,1(1,,1,,2)2(1(1,,2,,2)2,,1,,2)2)2}
699FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1+I) = ψ(ΩI+1II+1II+1+...))) = ψ(IIII...+1) = ψ(IεI+1+1)
ψ(M+χ(M+1)+ψχ(M)(M+χ(M+1)+1))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1(1,,2,,2)2,,1,,2)2)2}
700FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1I+1)) = ψ(ΩI+1+IIII...)
ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)))
{n,n(1,,1(1(1,,2,,2)2,,1,,2)3)2}
701FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1I+1+1))
ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)+1))
{n,n(1,,1(1(1,,2,,2)2,,1,,2)1,2)2}
702FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1I+1+I)) = ψ(IIII...+1) = ψ(IIεI+1+1)
ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)+χ(M)))
{n,n(1,,1(1(1,,2,,2)2,,1,,2)1(1,,1(1,,1,,2)1(1,,1,,2)2)2)2}
703FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1I+1ΩI+1I+1)))
ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)+ψχ(M+1)(M+χ(M+1))))
{n,n(1,,1(1(1,,2,,2)2,,1,,2)1(1(1,,2,,2)2,,1,,2)2)2}
704FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1I+1ΩI+1I+1ΩI+1I+1))))
ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)+ψχ(M+1)(M+χ(M+ψχ(M+1)(M+χ(M+1))))))
{n,n(1,,1(1(1(1,,2,,2)2,,1,,2)2(1,,2,,2)2,,1,,2)2)2}
705FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×2) = ψ(ΩI+1I+1) = ψ(ΩI+1ΩI+1I+1ΩI+1I+1ΩI+1I+1+...)))) = ψ(III...εI+1+1) = ψ(εI+2)
ψ(M+χ(M+1)×2) = ψ(M+χ(M+1)+ψχ(M+1)(M+χ(M+1)+ψχ(M+1)(M+χ(M+ψχ(M+1)(M+χ(M+...))))))
{n,n(1,,1(1(1,,2,,2)3,,1,,2)2)2}
706FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×3) = ψ(III...εI+2+1) = ψ(εI+3)
ψ(M+χ(M+1)×3)
{n,n(1,,1(1(1,,2,,2)3,,1,,2)2)2}
707FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×ω) = ψ(εI+ω)
ψ(M+χ(M+1)×ω)
{n,n(1,,1(1(1,,2,,2)1,2,,1,,2)2)2}
708FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×Ω) = ψ(εI+Ω)
ψ(M+χ(M+1)×Ω) = ψ(M+χ(M+1)×ψ(M+χ(M+1)×ψ(M+χ(M+1)×...)))
{n,n(1,,1(1(1,,2,,2)1'2,,1,,2)2)2}
709FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×I) = ψ(εI×2)
ψ(M+χ(M+1)×χ(M))
{n,n(1,,1(1(1,,2,,2)1(1,,1,,2)2,,1,,2)2)2}
710FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1×ψΩI+1I+1)) = ψ(ΩI+1×IIII...) = ψ(εεI+1)
ψ(M+χ(M+1)×ψχ(M+1)(M+χ(M+1)))
{n,n(1,,1(1(1,,2,,2)1(1(1,,2,,2)2,,1,,2)2,,1,,2)2)2}
711FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+12) = ψ(ΩI+1×ΩI+1) = ψ(ΩI+1×ψΩI+1I+1×ψΩI+1I+1×ψΩI+1I+1×...)))) = ψ(ζI+1)
ψ(M+χ(M+1)2)
{n,n(1,,1(1(1,,2,,2)1(1,,2,,2)2,,1,,2)2)2}
712FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ω) = ψ(φ(ω,I+1))
ψ(M+χ(M+1)ω)
{n,n(1,,1(1(1,,2,,2)2,,1,,2)2)2} = {n,n(2,,2,,2)2}
713FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1) = ψ(ГI+1) = ψ(φ(1,0,I+1))
ψ(M+χ(M+1)χ(M+1))
{n,n(1,,1(1(1(1,,2,,2)2,,2,,2)2,,1,,2)2)2}
714FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+12) = ψ(φ(1,0,0,I+1))
ψ(M+χ(M+1)χ(M+1)2)
{n,n(1,,1(1(1(1,,2,,2)3,,2,,2)2,,1,,2)2)2}
715FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1ω) = ψ(φ(1,0,0,...I+1))
ψ(M+χ(M+1)χ(M+1)ω)
{n,n(1,,1(1(1(1,,2,,2)1,2,,2,,2)2,,1,,2)2)2}
716FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+1ΩI+1ΩI+1)
ψ(M+χ(M+1)χ(M+1)χ(M+1))
{n,n(1,,1(1(1(1,,2,,2)1(1,,2,,2)2,,2,,2)2,,1,,2)2)2}
717FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI+2) = ψ(ΩI+1ΩI+1ΩI+1...) = ψ(εΩI+1+1)
ψ(2nd uncountable after 1st weakly inaccessible)
ψ(M+χ(M+2)) = ψ(M+χ(M+1)χ(M+1)χ(M+1)...)
{n,n(1,,1(1(1(1,,3,,2)2,,2,,2)2,,1,,2)2)2} = {n,n(1,,3,,2)2}
718FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI+3) = ψ(ΩI+2ΩI+2ΩI+2...) = ψ(εΩI+2+1)
ψ(3d uncountable after 1st weakly inaccessible)
ψ(M+χ(M+3)) = ψ(M+χ(M+2)χ(M+2)χ(M+2)...)
{n,n(1,,1(1(1(1(1,,4,,2)2,,3,,2)2,,2,,2)2,,1,,2)2)2} = {n,n(1,,4,,2)2}
719FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI+ω)
ψ(ω-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+ω))
{n,n(1,,1,2,,2)2}
720FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωII+ω))
ψ(M+χ(M+ω)+ψχ(M)(M+χ(M+ω)))
{n,n(1(1,,1(1,,1,2,,2)2)2,,1(1,,1,2,,2)2)2}
721FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωψII+ω)+1)
ψ(M+χ(M+ω)+Ωψχ(M)(M+χ(M+ω))+1)
{n,n(1,,2(1,,1,2,,2)2)2}
722FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωII+ω+1)) = ψ(ΩI+1ΩΩ...ψII+ω)+1)
ψ(M+χ(M+ω)+ψχ(M)(M+χ(M+ω)+1))
{n,n(1,,1(1,,1,,2)2(1,,1,2,,2)2)2}
723FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ω+I) = ψ(ΩI+ωII+ωII+ω+...)))
ψ(M+χ(M+ω)+χ(M))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1,,1,2,,2)2)2}
724FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+1)) = ψ(ΩI+ω+IIII...)
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+1)))
{n,n(1,,1(1,,2,,2)2(1,,1,2,,2)2)2}
725FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ω))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)))
{n,n(1,,1(1,,1,2,,2)3)2}
726FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ω+1))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+1))
{n,n(1,,1(1,,1,2,,2)1,2)2}
727FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ωΩI+1I+ω)))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω))))
{n,n(1,,1(1,,1,2,,2)1(1,,1,2,,2)2)2}
728FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ωΩI+1I+ω+1)))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+1)))
{n,n(1,,1(2,,1,2,,2)2)2}
729FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ωΩI+1I+ω+I)))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+χ(M))))
{n,n(1(1,,1,,2)2,,1,2,,2)2}
730FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+1I+ωΩI+1I+ωΩI+1I+ω+1))))
ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+1))))
{n,n(1,,1(2(1,,1,2,,2)2,,1,,2)2)2}
731FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωI+1) = ψ(ΩI+ωΩI+1I+ωΩI+1I+ωΩI+1I+ω+...))))
ψ(M+χ(M+ω)+χ(M+1)) = ψ(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+ψχ(M+1)(M+χ(M+ω)+...))))
{n,n(1(1,,2,,2)2(1,,1,2,,2)2,,1,,2)2}
732FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+2I+2))
ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+2)))
{n,n(1(1,,3,,2)2(1,,1,2,,2)2,,1,,2)2}
733FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+2I+ω))
ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ω)))
{n,n(1(1,,1,2,,2)3,,1,,2)2}
734FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+2I+ωΩI+2I+ω)))
ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ψχ(M+2)(M+χ(M+ω)))))
{n,n(1(1,,1,2,,2)1(1,,1,2,,2)2,,1,,2)2}
735FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+2I+ωΩI+2I+ω+1)))
ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ψχ(M+2)(M+χ(M+ω)+1))))
{n,n(1(2,,1,2,,2)2,,1,,2)2}
736FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+2I+ωΩI+2I+ωI+1)))
ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ψχ(M+2)(M+χ(M+ω)+χ(M+1)))))
{n,n(1(1,,2,,2)2,,1,2,,2)2}
737FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωI+2) = ψ(ΩI+ωΩI+2I+ωΩI+2I+ωΩI+2I+ω+....))))
ψ(M+χ(M+ω)+χ(M+2)) = ψ(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ψχ(M+2)(M+χ(M+ω)+ψχ(M+2)(M+χ(M+ω)+...)))))
{n,n(1(1,,3,,2)2(1,,1,2,,2)2,,1,,2)2}
738FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+3I+ωΩI+3I+ωI+2)))
ψ(M+χ(M+ω)+ψχ(M+3)(M+χ(M+ψχ(M+3)(M+χ(M+ω)+χ(M+2)))))
{n,n(1(1,,3,,2)2,,1,2,,2)2}
739FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωI+3) = ψ(ΩI+ωΩI+3I+ωΩI+3I+ωΩI+3I+ω+....))))
ψ(M+χ(M+ω)+χ(M+3)) = ψ(M+χ(M+ω)+ψχ(M+3)(M+χ(M+ψχ(M+3)(M+χ(M+ω)+ψχ(M+3)(M+χ(M+ω)+...)))))
{n,n(1(1,,4,,2)2(1,,1,2,,2)2,,1,,2)2}
740FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ω×2)
ψ(M+χ(M+ω)×2)
{n,n(1(1,,1,2,,2)2,,1,2,,2)2}
741FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ω2)
ψ(M+χ(M+ω)2)
{n,n(1(1,,1,2,,2)1(1,,1,2,,2)2,,1,2,,2)2}
742FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ωΩI+ω)
ψ(M+χ(M+ω)χ(M+ω))
{n,n(1(1(1,,1,2,,2)2,,1,2,,2)2,,1,2,,2)2}
743FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ω+1) = ψ(ΩI+ωΩI+ωΩI+ω...)
ψ(M+χ(M+ω+1)) = ψ(M+χ(M+ω)χ(M+ω)χ(M+ω)...)
{n,n(1,,2,2,,2)2}
744FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI+Ω) = ψ(ΩI+ψ(ΩI+ψ(ΩI+...)))
ψ(1st uncountable-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+Ω)) =ψ(M+χ(M+ψ(M+χ(M+ψ(M+χ(M+...))))))
{n,n(1,,1'2,,2)2}
745FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI+ψI(0))
ψ(1st fixed point of uncountable-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+ψχ(M)(M)))
{n,n(1,,1(1,,1(1,,1,,2)2)2,,2)2}
746FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ψI(I))
ψ(M+χ(M+ψχ(M)(M+χ(M))))
{n,n(1,,1(1,,1(1,,1,,2)1(1,,1,,2)2)2,,2)2}
747FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI+ψII+ω))
ψ(M+χ(M+ψχ(M)(M+χ(M+ω))))
{n,n(1,,1(1,,1(1,,1,2,,2)2)2,,2)2}
748FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI×2) = ψ(ΩI+I) = ψII+ψII+ψII+...))) = ψ(I(0,I(1,0)×2))
ψ(1st weakly inaccessible-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+χ(M))) = ψ(M+χ(M+ψχ(M)(M+χ(M+ψχ(M)(M+χ(M+...))))))
{n,n(1,,1(1,,1,,2)2,,2)2}
749FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI×2+1) = ψ(ΩI×2ΩI×2ΩI×2...)
ψ(M+χ(M+χ(M)+1))
{n,n(1,,2(1,,1,,2)2,,2)2}
750FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI×3)
ψ(M+χ(M+χ(M)+χ(M)))
{n,n(1,,1(1,,1,,2)3,,2)2}
751FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψΩI+1I+1)) = ψ(ΩIIII...)
ψ(M+χ(M+ψχ(M+1)(M+χ(M+1)))
{n,n(1,,1(1(1,,2,,2)2,,1,,2)2,,2)2}
752FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψΩI+1I×2))
ψ(M+χ(M+ψχ(M+1)(M+χ(M+χ(M))))
{n,n(1,,1(1(1,,1(1,,1,,2)2,,2)2,,1,,2)2,,2)2}
753FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψΩI+1ψΩI+1I+1))) = ψ(ΩψΩI+1IIII...))
ψ(M+χ(M+ψχ(M+1)(M+χ(M+ψχ(M+1)(M+χ(M+1))))
{n,n(1,,1(1(1,,1(1(1,,2,,2)2,,1,,2)2,,2)2,,1,,2)2,,2)2}
754FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩΩI+1) = ψ(ΩψΩI+1ψΩI+1ψΩI+1(...)))) = ψ(I(I(0,I(1,0)+1)))
ψ(1st uncountable after 1st weakly inaccessible-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+χ(M+1))) = ψ(M+χ ψ(M+χ(M+ψχ(M+1)(M+χ(M+ψχ(M+1)(M+χ(M+...))))
{n,n(1,,1(1,,2,,2)2,,2)2}
755FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩΩI×2) = ψ(I(I(0,I(1,0)×2)))
ψ(1st weakly inaccessible-th uncountable after 1st weakly inaccessible-th uncountable after 1st weakly inaccessible)
ψ(M+χ(M+χ(M+χ(M))))
{n,n(1,,1(1,,1(1,,1,,2)2,,2)2,,2)2}
756FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩΩI+1) = ψ(I(I(I(0,I(1,0)+1))))
ψ(M+χ(M+χ(M+χ(M+1))))
{n,n(1,,1(1,,1(2,,1,,2)2,,2)2,,2)2}
757FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩΩI×2) = ψ(I(I(I(0,I(1,0)×2))))
ψ(M+χ(M+χ(M+χ(M+χ(M)))))
{n,n(1,,1(1,,1(1,,1(1,,1,,2)2,,2)2,,2)2,,2)2}
758FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩΩΩI+1) = ψ(I(I(I(I(0,I(1,0)+1)))))
ψ(M+χ(M+χ(M+χ(M+χ(M+1)))))
{n,n(1,,1(1,,1(1,,1(2,,1,,2)2,,2)2,,2)2,,2)2}
759FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI2(0)) = ψΩαIΩI+αI2(0)))) = ψI(1,1)(0) = ψ(ΩΩΩ...ΩI+1) = ψ(I(0,I(0,I(0,I(0,...I(1,0)+1))))
ψ(1st fixed point of uncountable after 1st weakly inaccessible)
ψ(M×2) = ψ(M+M) = ψ(M+χ(M+χ(M+χ(M+χ(M+...)))))
{n,n(1,,1(1,,1,,3)2,,2)2} = {n,n(1,,1,,3)2}
760FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+1) = ψ(ψI2(0))×ω
ψ(M×2+1)
{n,n(2,,1(1,,1,,3)2,,2)2}
761FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+Ω) = εψII2(0))+1
ψ(M×2+Ω)
{n,n(1'2,,1(1,,1,,3)2,,2)2}
762FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψI(0))
ψ(M×2+ψχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1,,1,,3)2)2}
763FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψII+1))
ψ(M×2+ψχ(M)(M+χ(M+1)))
{n,n(1(1,,1(1,,2,,2)2)2,,1(1,,1,,3)2)2}
764FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψII×2))
ψ(M×2+ψχ(M)(M+χ(M+χ(M))))
{n,n(1(1,,1(1,,1(1,,1,,2)2,,2)2)2,,1(1,,1,,3)2)2}
765FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψII2(0)))
ψ(M×2+ψχ(M)(M+χ(M×2)))
{n,n(1(1,,1(1,,1,,3)2)2,,1(1,,1,,3)2)2}
766FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ΩψII2(0))+1)
ψ(M×2+Ωψχ(M)(M+χ(M×2))+1)
{n,n(1,,2(1,,1,,3)2)2}
767FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψII2(0)+1)) = ψ(ψI2(0)+ΩΩΩ...ψII2(0))+1)
ψ(M×2+ψχ(M)(M+χ(M×2)+1))
{n,n(1,,1(1,,1,,2)2(1,,1,,3)2)2}
768FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+I) = ψ(ψI2(0)+ψII2(0)+ψII2(0)+...)))
ψ(M×2+χ(M)) = ψ(M×2+ψχ(M)(M+χ(M×2)+ψχ(M)(M+χ(M×2)+ψχ(M)(M+χ(M×2)+...))))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1,,1,,3)2)2}
769FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I+1)) = ψ(ψI2(0)+IIII...)
ψ(M×2+ψχ(M+1)(M+χ(M+1))
{n,n(1,,1(1,,2,,2)2(1,,1,,3)2)2}
770FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I×2))
ψ(M×2+ψχ(M+1)(M+χ(M+χ(M)))
{n,n(1,,1(1,,1(1,,1,,2)2,,2)2(1,,1,,3)2)2}
771FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I2(0)))
ψ(M×2+ψχ(M+1)(M+χ(M×2)))
{n,n(1,,1(1,,1,,3)3)2}
772FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I2(0)+ψΩI+1I2(0))))
ψ(M×2+ψχ(M+1)(M+χ(M×2+ψχ(M+1)(M+χ(M×2)))))
{n,n(1,,1(1,,1,,3)1(1,,1,,3)2)2}
773FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I2(0)+ψΩI+1I2(0)+1)))
ψ(M×2+ψχ(M+1)(M+χ(M×2+ψχ(M+1)(M+χ(M×2))+1)))
{n,n(1,,1(2,,1(1,,1,,3)2,,2)2)2}
774FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+1I2(0)+ψΩI+1I2(0)+I)))
ψ(M×2+ψχ(M+1)(M+χ(M×2+ψχ(M+1)(M+χ(M×2))+χ(M))))
{n,n(1(1,,1,,2)2,,1(1,,1,,3)2,,2)2}
775FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+2I2(0)+ψΩI+2I2(0)+I)))
ψ(M×2+ψχ(M+2)(M+χ(M×2+ψχ(M+2)(M+χ(M×2))+χ(M))))
{n,n(1(1,,2,,2)2,,1(1,,1,,3)2,,2)2}
776FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ψΩI+3I2(0)+ψΩI+3I2(0)+I)))
ψ(M×2+ψχ(M+3)(M+χ(M×2+ψχ(M+3)(M+χ(M×2))+χ(M))))
{n,n(1(1,,3,,2)2,,1(1,,1,,3)2,,2)2}
777FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ΩI+ω)
ψ(M×2+χ(M+ω))
{n,n(1(1,,1,2,,2)2,,1(1,,1,,3)2,,2)2}
778FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)+ΩI×2)
ψ(M×2+χ(M+χ(M)))
{n,n(1(1,,1(1,,1,,2)2,,2)2,,1(1,,1,,3)2,,2)2}
779FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(0)×2) = ψ(ψI2(0)+ψI2(0)) = ψ(ψI2(0)+ΩΩΩ...ΩI+1)
ψ(M×2+ψχ(M×2)(M×2))
{n,n(1(1,,1(1,,1,,3)2,,2)2,,1(1,,1,,3)2,,2)2}
780FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI2(0)+1)
ψ(M×2+Ωψχ(M×2)(M×2)+1)
{n,n(1,,2(1,,1,,3)2,,2)2}
781FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI2(0)+I)
ψ(M×2+Ωψχ(M×2)(M×2)+χ(M))
{n,n(1,,1(1,,1,,2)2(1,,1,,3)2,,2)2}
782FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI2(0)+ΩI+1)
ψ(M×2+Ωψχ(M×2)(M×2)+χ(M+χ(M)))
{n,n(1,,1(1,,2,,2)2(1,,1,,3)2,,2)2}
783FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI2(0)×2)
ψ(M×2+Ωψχ(M×2)(M×2)×2)
{n,n(1,,1(1,,1(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2}
784FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩψI2(0)+1) = ψ(ΩψΩψI2(0)+1ψΩψI2(0)+1ψΩψI2(0)+1(...)))) = ψ(ΩΩΩ...ΩI+1+1)
ψ(M×2+ΩΩψχ(M×2)(M×2)+1)
{n,n(1,,1(1,,2(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2}
785FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩψI2(0)×2)
ψ(M×2+ΩΩψχ(M×2)(M×2)×2)
{n,n(1,,1(1,,1(1,,1(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2}
786FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩΩψI2(0)+1) = ψ(ΩΩΩ...ΩI+1+1)
ψ(M×2+ΩΩΩψχ(M×2)(M×2)+1)
{n,n(1,,1(1,,1(1,,2(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2}
787FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(1)) = ψ(ψI2(0)+ΩΩΩ...ψI2(0)+1)
ψ(M×2+ψχ(M×2)(M×2+1)) = ψ(M×2+ΩΩΩ...ψχ(M×2)(M×2)+1)
{n,n(1,,1(1,,1,,3)3,,2)2}
788FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(Ω)) = ψ(ψI2(ψ(ψI2(ψ(ψI2(...))))))
ψ(M×2+ψχ(M×2)(M×2+Ω))
{n,n(1,,1(1,,1,,3)1'2,,2)2}
789FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2(I)) = ψ(ψI2II2II2(...))))))
ψ(M×2+ψχ(M×2)(M×2+χ(M)))
{n,n(1,,1(1,,1,,3)1(1,,1,,2)2,,2)2}
790FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2I+1)) = ψ(ψI2ΩI+1I2ΩI+1I2(...))))))
ψ(M×2+ψχ(M×2)(M×2+χ(M+1)))
{n,n(1,,1(1,,1,,3)1(1,,2,,2)2,,2)2}
791FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2I×2))
ψ(M×2+ψχ(M×2)(M×2+χ(M+χ(M))))
{n,n(1,,1(1,,1,,3)1(1,,1(1,,1,,2)2,,2)2,,2)2}
792FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2I2(0))) = ψ(ψI2ΩΩ...ΩI+1))
ψ(M×2+ψχ(M×2)(M×2+ψχ(M×2)(M×2)))
{n,n(1,,1(1,,1,,3)1(1,,1(1,,1,,3)2,,2)2,,2)2}
793FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI2I2I2(0))))
ψ(M×2+ψχ(M×2)(M×2+ψχ(M×2)(M×2+ψχ(M×2)(M×2))))
{n,n(1,,1(1,,1,,3)1(1,,1(1,,1,,3)1(1,,1(1,,1,,3)2,,2)2,,2)2,,2)2}
794FGH ψ-function
FGH ψ(ψM)
ψ(cardinal)

SAN
ψ(I2) = ψΩαIΩI+αI2(I2)))) = ψ(ψI2I2I2I2(...))))) = ψ(I(1,1)) =
ψ(M×2+χ(M×2)) = ψ(M×2+ψχ(M×2)(M×2+ψχ(M×2)(M×2+ψχ(M×2)(M×2+...))))
ψ(2nd weakly inaccessible)
{n,n(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
795FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2+1) = ψ(I2)×ω = ψ(ψI2I2I2I2(...))))+1)
ψ(M×2+χ(M×2)+1)
{n,n(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
796FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I(I2))
ψ(M×2+χ(M×2)+ψχ(M)(M×2+χ(M×2)))
{n,n(1(1,,1(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2)2,,1(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2)2}
797FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2ψI(I2)+1)
ψ(M×2+χ(M×2)+Ωψχ(M)(M×2+χ(M×2))+1)
{n,n(1,,2(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2)2}
798FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I(I2+1)) = ψ(ψI2(0)+ΩΩΩ...ψI(I2)+1)
ψ(M×2+χ(M×2)+ψχ(M)(M×2+χ(M×2)+1))
{n,n(1,,1(1,,1,,2)2(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2)2}
799FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I+1)
ψ(M×2+χ(M×2)+χ(M+1))
{n,n(1,,1(1,,2,,2)2(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2)2}
800FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I+ω)
ψ(M×2+χ(M×2)+χ(M+ω))
{n,n(1(1,,1,2,,2)2,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
801FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2(0))
ψ(M×2+χ(M×2)+ψχ(M×2)(M×2))
{n,n(1(1,,1(1,,1,,3)2,,2)2,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
802FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2I2(0)))
ψ(M×2+χ(M×2)+ψχ(M×2)(M×2+ψχ(M×2)(M×2)))
{n,n(1(1,,1(1,,1,,3)1(1,,1(1,,1,,3)2,,2)2,,2)2,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
803FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2(I2))
ψ(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2)))
{n,n(1(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2,,1(1,,1,,3)1(1,,1,,3)2,,2)2}
804FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2ψI2(I2)+1)=ψ(I2ψI2(I2)+1)
ψ(M×2+χ(M×2)+Ωψχ(M×2)(M×2+χ(M×2))+1)
{n,n(1,,2(1,,1,,3)1(1,,1,,3)2,,2)2}
805FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2ψI2(I2)×2)
ψ(M×2+χ(M×2)+Ωψχ(M×2)(M×2+χ(M×2))×2)
{n,n(1,,1(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2(1,,1,,3)1(1,,1,,3)2,,2)2}
806FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2(I2+1)) = ψ(I2ΩΩ...ψI2(I2)+1) = ψ(ψI2I2I2I2(...)))+1))
ψ(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2)+1))
{n,n(1,,1(1,,1,,3)2(1,,1,,3)2,,2)2}
807FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2(I2I2(I2)))
ψ(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2))))
{n,n(1,,1(1,,1,,3)1(1,,1(1,,1,,3)1(1,,1,,3)2,,2)2(1,,1,,3)2,,2)2}
808FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2×2) = ψ(I2I2(I2I2(I2I2(...)))) = ψ(ψI2I2I2I2(...ψI2(I2)+1...)))))
ψ(M×2+χ(M×2)+χ(M×2)) = ψ(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2)+ψχ(M×2)(M×2+χ(M×2)+...))))
{n,n(1,,1(1,,1,,3)1(1,,1,,3)3,,2)2}
809FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I22)
ψ(M×2+χ(M×2)2)
{n,n(1,,1(1,,1,,3)1(1,,1,,3)1(1,,1,,3)2,,2)2}
810FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I2I2)
ψ(M×2+χ(M×2)χ(M×2))
{n,n(1,,1(1(1,,1,,3)2,,1,,3)2,,2)2}
811FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI2+1) = ψ(I2I2I2...)
ψ(1st uncountable after 2nd weakly inaccessible)
ψ(M×2+χ(M×2+1)) = ψ(M×2+χ(M×2)χ(M×2)χ(M×2)...)
{n,n(1,,1(1(1,,2,,3)2,,1,,3)2,,2)2} = {n,n(1,,2,,3)2}
812FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI2)
ψ(M×2+χ(M×2+ω))
{n,n(1,,1,2,,3)2}
813FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI2+I)
ψ(M×2+χ(M×2+χ(M)))
{n,n(1,,1(1,,1,,2)2,,3)2}
814FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI2I2(0))
ψ(M×2+χ(M×2+ψχ(M×2)(M×2))
{n,n(1,,1(1,,1(1,,1,,3)2,,2)2,,3)2}
815FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI2I2I2+I))
ψ(M×2+χ(M×2+ψχ(M×2)(M×2+χ(M)))
{n,n(1,,1(1,,1(1,,1(1,,1,,2)2,,3)2,,2)2,,3)2}
816FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI2×2) = ψ(ΩI2I2I2+ψI2I2+...)))
ψ(M×2+χ(M×2+χ(M×2))) = ψ(M×2+χ(M×2+ψχ(M×2)(M×2+χ(M×2+ψχ(M×2)(M×2+χ(M×2+...)))))
{n,n(1,,1(1,,1,,3)2,,3)2}
817FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩI2+1) =
ψ(M×2+χ(M×2+χ(M×2+1)))
{n,n(1,,1(1,,2,,3)2,,3)2}
818FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩI2×2)
ψ(M×2+χ(M×2+χ(M×2+χ(M×2))))
{n,n(1,,1(1,,1(1,,1,,3)2,,3)2,,3)2}
819FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩΩI2+1)
ψ(M×2+χ(M×2+χ(M×2+χ(M×2+1))))
{n,n(1,,1(1,,1(1,,2,,3)2,,3)2,,3)2}
820FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI3(0)) = ψΩαIΩI+αI2ΩI2I3(0)))))) = ψI(1,2)(0) = ψ(ΩΩΩ...ΩI2+1) = ψ(I(0,I(0,I(0,I(0,...I(1,1)+1))))
ψ(1st fixed point uncountable after 2nd weakly inaccessible)
ψ(M×3) = ψ(M×2+M) = ψ(M×2+χ(M×2+χ(M×2+χ(M×2+...))))
{n,n(1,,1(1,,1,,4)2,,3)2} = {n,n(1,,1,,4)2}
821FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI3(0)+1)
ψ(M×3+Ωψχ(M×3)(M×3)+1)
{n,n(1,,2(1,,1,,4)2,,3)2}
822FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI3(0)×2)
ψ(M×3+Ωψχ(M×3)(M×3)×2)
{n,n(1,,1(1,,1(1,,1,,4)2,,3)2(1,,1,,4)2,,3)2}
823FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI3(1)) = ψ(ψI2(0)+ΩΩΩ...ψI3(0)+1)
ψ(M×3+ψχ(M×3)(M×3+1))
{n,n(1,,1(1,,1,,4)3,,3)2}
824FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI3I3(0)))
ψ(M×3+ψχ(M×3)(M×3+ψχ(M×3)(M×3)))
{n,n(1,,1(1,,1,,4)1(1,,1(1,,1,,4)2,,3)2,,3)2}
825FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I3) = ψΩαIΩI+αI2ΩI2I3(I3)))))) = ψ(ψI3I3I3I3(...))))) = ψ(I(1,2))
ψ(3d weakly inaccessible)
ψ(M×3+χ(M×3)) = ψ(M×3+ψχ(M×3)(M×3+ψχ(M×3)(M×3+ψχ(M×3)(M×3+...))))
{n,n(1,,1(1,,1,,4)1(1,,1,,4)2,,3)2}
826FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I3I3(I3))
ψ(M×3+χ(M×3)+ψχ(M×3)(M×3))
{n,n(1(1,,1(1,,1,,4)1(1,,1,,4)2,,3)2,,1(1,,1,,4)1(1,,1,,4)2,,3)2}
827FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I3I3(I3+1))
ψ(M×3+χ(M×3)+ψχ(M×3)(M×3+1))
{n,n(1,,1(1,,1,,4)2(1,,1,,4)2,,3)2}
828FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I3×2)
ψ(M×3+χ(M×3)×2)
{n,n(1,,1(1,,1,,4)1(1,,1,,4)3,,3)2}
829FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI3+1)
ψ(M×3+χ(M×3+1))
{n,n(1,,2,,4)2}
830FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI3×2)
ψ(M×3+χ(M×3+χ(M×3)))
{n,n(1,,1(1,,1,,4)2,,4)2}
831FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI4(0)) = ψI(1,3)(0) = ψ(ΩΩΩ...ΩI3+1) = ψ(I(0,I(0,I(0,I(0,...I(1,2)+1))))
ψ(1st fixed point uncountable after 3d weakly inaccessible)
ψ(M×4) = ψ(M×3+M) = ψ(M×3+χ(M×3+χ(M×3+χ(M×3+...))))
{n,n(1,,1(1,,1,,5)2,,4) = {n,n(1,,1,,5)2}
832FGH ψ-function
FGH ψ(ψM)
ψ(cardinal)

SAN
ψ(I4) = ψ(ψI4I4I4I4(...))))) = ψ(I(1,3))
ψ(M×4+χ(M×4))
ψ(4-th weakly inaccessible)
{n,n(1,,1(1,,1,,5)1(1,,1,,5)2,,4)2}
833FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI5(0)) = ψI(1,4)(0) = ψ(ΩΩΩ...ΩI4+1) = ψ(I(0,I(0,I(0,I(0,...I(1,3)+1))))
ψ(1st fixed point uncountable after 4-th weakly inaccessible)
ψ(M×5) = ψ(M×4+M) = ψ(M×4+χ(M×4+χ(M×4+χ(M×4+...))))
{n,n(1,,1(1,,1,,6)2,,5)2} = {n,n(1,,1,,6)2}
834FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I5) = ψ(ψI4I4I4I4(...))))) = ψ(I(1,4))
ψ(5-th weakly inaccessible)
ψ(M×5+χ(M×5))
{n,n(1,,1(1,,1,,5)1(1,,1,,5)2,,4)2}
835FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(Iω) = ψ(Isup(n)) = ψΩαIΩI+αI2ΩI2I3ΩI3I4(...))))))))
ψ(ω-th weakly inaccessible)
ψ(M×ω)
{n,n(1,,1,,1,2)2}
836FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI(0))
ψ(M×ω+ψχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1,,1,,1,2)2)2}
837FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI(Iω))
ψ(M×ω+ψχ(M)(M×ω))
{n,n(1(1,,1(1,,1,,1,2)2)2,,1(1,,1,,1,2)2)2}
838FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωψI(Iω)+1)
ψ(M×ω+Ωψχ(M)(M×ω)+1)
{n,n(1,,2(1,,1,,1,2)2)2}
839FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI(Iω+1)) = ψ(IωΩΩ...ψI(Iω)+1)
ψ(M×ω+ψχ(M)(M×ω+1))
{n,n(1,,1(1,,1,,2)2(1,,1,,1,2)2)2}
840FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω+I) = ψ(IωI(IωI(Iω+...)))
ψ(M×ω+χ(M))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1,,1,,1,2)2)2}
841FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI+1I+1)) = ψ(Iω+ IIII...)
ψ(M×ω+ψχ(M+1)(M+1))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1,,1,,1,2)2)2}
842FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI+1(Iω))
ψ(M×ω+ψχ(M+1)(M×ω))
{n,n(1,,1(1,,1,,1,2)3)2}
843FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI+1(IωΩI+1(Iω)))
ψ(M×ω+ψχ(M+1)(M×ω+ψχ(M+1)(M×ω)))
{n,n(1,,1(1,,1,,1,2)1(1,,1,,1,2)2)2}
844FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI+1(IωΩI+1(Iω+1)))
ψ(M×ω+ψχ(M+1)(M×ω+ψχ(M+1)(M×ω+1)))
{n,n(1,,1(2,,1,,1,2)2)2}
845FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI+1(IωΩI+1(Iω+I)))
ψ(M×ω+ψχ(M+1)(M×ω+ψχ(M+1)(M×ω+χ(M))))
{n,n(1(1,,1,,2)2,,1,,1,2)2}
846FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI+1) = ψ(IωΩI+1(IωΩI+1(IωΩI+1(Iω+...))))
ψ(M×ω+χ(M+1)) = ψ(M×ω+ψχ(M+1)(M×ω+ψχ(M+1)(M×ω+ψχ(M+1)(M×ω+...))))
{n,n(1(1,,2,,2)1(1,,1(1,,1,,1,2)2,,2)2,,1,,2)2}
847FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI+ω)
ψ(M×ω+χ(M+ω))
{n,n(1(1,,1,2,,2)2,,1(1,,1,,1,2)2,,2)2}
848FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI2(0))
ψ(M×ω+ψχ(M+2)(M))
{n,n(1(1,,1(1,,1,,3)2,,2)2,,1(1,,1,,1,2)2,,2)2}
849FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI2(Iω))
ψ(M×ω+ψχ(M+2)(M×ω))
{n,n(1(1,,1(1,,1,,1,2)2,,2)2,,1(1,,1,,1,2)2,,2)2}
850FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωψI2(Iω)+1)
ψ(M×ω+Ωψχ(M+2)(M×ω)+1)
{n,n(1,,2(1,,1,,1,2)2,,2)2}
851FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωI2(Iω+1)) = ψ(IωΩΩ...ψI2(Iω)+1)
ψ(M×ω+ψχ(M+2)(M×ω+1))
{n,n(1,,1(1,,1,,2)2(1,,1,,1,2)2)2}
852FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω+I2) = ψ(IωI2(IωI2(Iω+...)))
ψ(M×ω+χ(M×2))
{n,n(1,,1(1,,1,,3)1(1,,1,,3)2(1,,1,,1,2)2,,2)2}
853FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI2+1(Iω))
ψ(M×ω+ψχ(M×2+1)(M×ω))
{n,n(1,,1(1,,1,,1,2)3,,2)2}
854FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI2+1(IωΩI2+1(Iω)))
ψ(M×ω+ψχ(M×2+1)(M×ω+ψχ(M×2+1)(M×ω)))
{n,n(1,,1(1,,1,,1,2)1(1,,1,,1,2)2,,2)2}
855FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI2+1(IωΩI2+1(Iω+1)))
ψ(M×ω+ψχ(M×2+1)(M×ω+ψχ(M×2+1)(M×ω+1)))
{n,n(1,,1(2,,1,,1,2)2,,2)2}
856FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI2+1(IωΩI2+1(Iω+I2)))
ψ(M×ω+ψχ(M×2+1)(M×ω+ψχ(M×2+1)(M×ω+χ(M×2))))
{n,n(1(1,,1,,3)2,,1,,1,2)2}
857FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI3+1(IωΩI3+1(Iω+I3)))
ψ(M×ω+ψχ(M×3+1)(M×ω+ψχ(M×3+1)(M×ω+χ(M×3))))
{n,n(1(1,,1,,4)2,,1,,1,2)2}
858FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IωΩI4+1(IωΩI4+1(Iω+I4)))
ψ(M×ω+ψχ(M×4+1)(M×ω+ψχ(M×4+1)(M×ω+χ(M×4))))
{n,n(1(1,,1,,5)2,,1,,1,2)2}
859FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω×2) = ψ(Iω+Iω)
ψ(M×ω+χ(M×ω))
{n,n(1(1,,1,,1,2)2,,1,,1,2)2}
860FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩIω+1)
ψ(M×ω+χ(M×ω+1))
{n,n(1,,2,,1,2)2}
861FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩIω+I)
ψ(M×ω+χ(M×ω+χ(M)))
{n,n(1,,1(1,,1,,2)2,,1,2)2}
862FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩIω×2)
ψ(M×ω+χ(M×ω+χ(M×ω)))
{n,n(1,,1(1,,1,,1,2)2,,1,2)2}
863FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψIω+1(0))
ψ(M×(ω+1)) = ψ(M×ω+M)
{n,n(1,,1,,2,2)2}
864FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω+1)
ψ(M×(ω+1)+χ(M×(ω+1)))
{n,n(1,,1(1,,1,,2,2)1(1,,1,,2,2)2,,1,2)2}
865FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψIω+2(0))
ψ(M×(ω+2))
{n,n(1,,1,,3,2)2}
866FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω+2)
ψ(M×(ω+2)+χ(M×(ω+2)))
{n,n(1,,1(1,,1,,3,2)1(1,,1,,3,2)2,,2,2)2}
867FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iω×2) = ψ(Isup(ω+n))
ψ(M×(ω×2))
{n,n(1,,1,,1,3)2}
868FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩIω×2+1)
ψ(M×(ω×2)+χ(M×(ω×2)))
{n,n(1,,2,,1,3)2}
869FGH ψ-function
FGH ψ(ψM)
SAN
ψ(Iε0) = ψ(Iψ(Ω)))
ψ(M×ψ(Ω))
{n,n(1,,1,,1(1'2)2)2}
870FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(IΩ) = ψ(Iψ(Iψ(Iψ(...))))) = ψ(I(1,I(0,0))
ψ(1st uncountable ordinal-th weakly inaccessible)
ψ(M×Ω) = ψ(M×ψ(M×ψ(M×ψ(M×...))))
{n,n(1,,1,,1'2)2}
871FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(IψI(0)) = ψ(IΩΩΩ... )
ψ(1st fixed point of uncountable ordinal-th weakly inaccessible)
ψ(M×ψχ(M)(M))
{n,n(1,,1,,1(1,,1(1,,1,,2)2)2)2}
872FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IψI(IψI(0)))
ψ(M×ψχ(M)(M×ψχ(M)(M)))
{n,n(1,,1,,1(1,,1(1,,1,,1(1,,1(1,,1,,2)2)2)2)2)2}
873FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(II) = ψ(IψI(IψI(IψI(...))))) = ψ(I(1,I(1,0)))
ψ(1st weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M)) = ψ(M×ψχ(M)(M×ψχ(M)(M×ψχ(M)(M×...))))
{n,n(1,,1,,1(1,,1,,2)2)2}
874FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩII+1)
ψ(M×Ωχ(M)+1)
{n,n(1,,2,,1(1,,1,,2)2)2}
875FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψII+1(0)) = ψ(ΩΩΩ...ΩII+1)
ψ(M×ψχ(M×χ(M)+1)(M×χ(M)+1))
{n,n(1,,1,,2(1,,1,,2)2)2}
876FGH ψ-function
FGH ψ(ψM)
SAN
ψ(II+1)
ψ(M×χ(M)+1)
{n,n(1,,1,,2(1,,1,,2)1(1,,1,,2)2)2}
877FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψII+2(0)) = ψ(ΩΩΩ...ΩII+1+1)
ψ(M×ψχ(M×χ(M)+2)(M×χ(M)+2))
{n,n(1,,1,,3(1,,1,,2)2)2}
878FGH ψ-function
FGH ψ(ψM)
SAN
ψ(II+2)
ψ(M×χ(M)+2)
{n,n(1,,1,,3(1,,1,,2)1(1,,1,,2)2)2}
879FGH ψ-function
FGH ψ(ψM)
SAN
ψ(II+ω)
ψ(M×χ(M)+ω)
{n,n(1,,1,,1,2(1,,1,,2)2)2}
880FGH ψ-function
FGH ψ(ψM)
SAN
ψ(II×2)
ψ(M×χ(M)+χ(M)))
{n,n(1,,1,,1(1,,1,,2)3)2}
881FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(IΩI+1)
ψ(1st uncountable after 1st weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M+1)))
{n,n(1,,1,,1(1,,2,,2)2)2}
882FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IψI2(Iω))
ψ(M×ψχ(M×2)(M×ω))
{n,n(1,,1,,1(1,,1(1,,1,,1,2)2,,2)2)2}
883FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(II2) = ψ(IψI2(IψI2(IψI2(...))))) = ψ(I(1,I(1,1)))
ψ(2nd weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M×2))
{n,n(1,,1,,1(1,,1,,3)2)2}
884FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(II3) = ψ(I(1,I(1,2)))
ψ(3d weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M×3))
{n,n(1,,1,,1(1,,1,,4)2)2}
885FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(IIω) = ψ(I(1,I(1,ω)))
ψ(ω-th weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M×ω))
{n,n(1,,1,,1(1,,1,,1,2)2)2}
886FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(IIIω) = ψ(I(1,I(1,I(1,ω))))
ψ(ω-th weakly inaccessible-th weakly inaccessible-th weakly inaccessible)
ψ(M×χ(M×χ(M×ω)))
{n,n(1,,1,,1(1,,1,,1(1,,1,,1,2)2)2)2}
887FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)) = ψ(IIII...) = ψ(I(1,I(1,I(1,I(1,...)))))
ψ(1st fixed point of weakly inaccessible)
ψ(M2) = ψ(M×M) = ψ(ψχ(M2)(M2))
{n,n(1,,1,,1(1,,1,,1,,2)2)2} = {n,n(1,,1,,1,,2)2}
888FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+ψII(2,0)(0)))
ψ(M2χ(M)(M2))
{n,n(1(1,,1(1,,1,,1(1,,1,,1,,2)2)2)2,,1(1,,1,,1(1,,1,,1,,2)2)2)2}
889FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+ψΩI+1I(2,0)(0)))
ψ(M2χ(M+1)(M2))
{n,n(1,,1(1,,1,,1(1,,1,,1,,2)2)3)2}
890FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+ψΩI+1I(2,0)(0)+ψΩI+1I(2,0)(0)+I)))
ψ(M2χ(M+1)(M2χ(M+1)(M2+χ(M))))
{n,n(1(1,,1,,2)2,,1,,1(1,,1,,1,,2)2)2}
891FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+ψΩI2+1I(2,0)(0)+ψΩI2+1I(2,0)(0)+I2)))
ψ(M2χ(M×2+1)(M2χ(M×2+1)(M2+χ(M×2))))
{n,n(1(1,,1,,3)2,,1,,1(1,,1,,1,,2)2)2}
892FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+ψΩI3+1I(2,0)(0)+ψΩI3+1I(2,0)(0)+I3)))
ψ(M2χ(M×3+1)(M2χ(M×3+1)(M2+χ(M×3))))
{n,n(1(1,,1,,4)2,,1,,1(1,,1,,1,,2)2)2}
893FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)+Iω)
ψ(M2+χ(M×ω))
{n,n(1(1,,1,,1,2)2,,1,,1(1,,1,,1,,2)2)2}
894FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(0)×2) = ψ(ψI(2,0)(0)+IIII...)
ψ(M2χ(M2)(M2)) = ψ(M2+χ(M×ψχ(M2)(M2)))
{n,n(1(1,,1,,1(1,,1,,1,,2)2)2,,1,,1(1,,1,,1,,2)2)2}
895FGH ψ-function
FGH ψ(ψM)
ψ(cardinal)

SAN
ψ(ΩψI(2,0)(0)+1)
ψ(M2ψχ(M2)(M2)+1)
ψ(1st uncountable after 1st fixed point of weakly inaccessible)
{n,n(1,,2,,1(1,,1,,1,,2)2)2}
896FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI(2,0)(0)×2)
ψ(M2ψχ(M2)(M2)×2)
{n,n(1,,1(1,,1,,1(1,,1,,1,,2)2)2,,1(1,,1,,1,,2)2)2}
897FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(2,0)(0)+1)(0)) = ψ(ψIψI(2,0)(0)+1)(0)) = ψ(ΩΩΩ...ψI(2,0)(0)+1)
ψ(1st fixed point of uncountable after 1st fixed point of weakly inaccessible)
ψ(M2χ(M×(ψχ(M2)(M2)+1))(M×(ψχ(M2)(M2)+1)))
{n,n(1,,1(1,,1,,2(1,,1,,1,,2)2)2,,1(1,,1,,1,,2)2)2} = {n,n(1,,1,,2(1,,1,,1,,2)2)2}
898FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(2,0)(0)+1)(1))
ψ(2nd fixed point of uncountable after 1st fixed point of weakly inaccessible)
ψ(M2χ(M×(ψχ(M2)(M2)+1))(M×(ψχ(M2)(M2)+1)+1))
{n,n(1,,1(1,,1,,2(1,,1,,1,,2)2)3,,1(1,,1,,1,,2)2)2}
899FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,ψI(2,0)(0)+1)) = ψ(IψI(2,0)(0)+1)
ψ(1st weakly inaccessible after 1st fixed point of weakly inaccessible)
ψ(M2+χ(M×ψχ(M2)(M2)+1))
{n,n(1,,1(1,,1,,2(1,,1,,1,,2)2)1(1,,1,,2(1,,1,,1,,2)2)2,,1(1,,1,,1,,2)2)2}
900FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI(1,ψI(2,0)(0)+1)+1) = ψ(ΩIψI(2,0)(0)+1+1)
ψ(1st uncountable after 1st weakly inaccessible after 1st fixed point of weakly inaccessible)
ψ(M2χ(M×ψχ(M2)(M2)+1)+1)
{n,n(1,,2,,2(1,,1,,1,,2)2)2}
901FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(2,0)(0)+2)(0)) = ψ(ψIψI(2,0)(0)+2)(0)) = ψ(ΩΩΩ...IψI(2,0)(0)+1+1)
ψ(1st fixed point of uncountable after 1st weakly inaccessible after 1st fixed point of weakly inaccessible)
ψ(M2χ(M×(ψχ(M2)(M2)+2))(M×(ψχ(M2)(M2)+2)))
{n,n(1,,1,,3(1,,1,,1,,2)2)2}
902FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(2,0)(0)+3)(0)) = ψ(ψIψI(2,0)(0)+3)(0)) = ψ(ΩΩΩ...IψI(2,0)(0)+2+1)
ψ(1st fixed point of uncountable after 2nd weakly inaccessible after 1st fixed point of weakly inaccessible)
ψ(M2χ(M×(ψχ(M2)(M2)+3))(M×(ψχ(M2)(M2)+3)))
{n,n(1,,1,,3(1,,1,,1,,2)2)2}
903FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,ψI(2,0)(0)+ω))
ψ(M2χ(M×(ψχ(M2)(M2)+ω))(M×(ψχ(M2)(M2)+ω)))
{n,n(1,,1,,1,2(1,,1,,1,,2)2)2}
904FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,ψI(2,0)(0)×2))
ψ(M2+χ(M×(ψχ(M2)(M2)×2)))
{n,n(1,,1,,1(1,,1,,2(1,,1,,1,,2)2)2(1,,1,,1,,2)2)2}
905FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(1,ψI(2,0)(0)+1)))
ψ(M2+χ(M×χ(M×(ψχ(M2)(M2)+1))))
{n,n(1,,1(1,,1,,1(1,,1,,2(1,,1,,1,,2)2)2(1,,1,,1,,2)2)1(1,,1,,2(1,,1,,1,,2)2)2,,1(1,,1,,1,,2)2)2}
906FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(1,ψI(2,0)(0)×2)))
ψ(M2+χ(M×χ(M×(ψχ(M2)(M2)×2))))
{n,n(1,,1,,1(1,,1,,1(1,,1,,2(1,,1,,1,,2)2)2(1,,1,,1,,2)2)2(1,,1,,1,,2)2)2}
907FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(1)) = ψ(I(1,I(1,I(1,I(1,...ψI(2,0)(0)+1...)))) = ψ(M2+1)
ψ(2nd fixed point of weakly inaccessible)
ψ(M2χ(M2)(M2+1)) = ψ(M2+χ(M×χ(M×χ(M×...(ψχ(M2)(M2)+1)...))))
{n,n(1,,1,,1(1,,1,,1,,2)3)2}
908FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(I))
ψ(1st weakly inaccessible-th fixed point of weakly inaccessible)
ψ(M2χ(M2)(M2+χ(M)))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,2)2)2}
909FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)(I2))
ψ(2nd weakly inaccessible-th fixed point of weakly inaccessible)
ψ(M2χ(M2)(M2+χ(M×2)))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,3)2)2}
910FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)I(2,0)(0))) = ψ(ψI(2,0)(IIII...))
ψ(1st fixed point of weakly inaccessible-th fixed point of weakly inaccessible)
ψ(M2χ(M2)(M2χ(M2)(M2)))) = ψ(M2χ(M2)(M2+χ(M×χ(M×χ(M×...)))))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,1(1,,1,,1,,2)2)2)2}
911FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0)I(2,0)(1))) = ψ(ψI(2,0)(ψ(I(1,I(1,I(1,I(1,...ψI(2,0)(0)+1...))))))
ψ(1st fixed point of weakly inaccessible-th fixed point of weakly inaccessible)
ψ(M2χ(M2)(M2χ(M2)(M2+1))) = ψ(M2χ(M2)(M2+χ(M×χ(M×χ(M×...(ψχ(M2)(M2)+1)...)))))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,1(1,,1,,1,,2)3)2)2}
912FGH ψ-function
ψ(cardinal)

FGH ψ(ψM)
SAN
ψ(I(2,0)) = ψ(ψI(2,0)(I(2,0))) = ψ(ψI(2,0)I(2,0)I(2,0)I(2,0)(...)))))
ψ(1st 2-weakly inaccessible)
ψ(M2+χ(M2)) = ψ(M2χ(M2)(M2χ(M2)(M2χ(M2)(M2+...))))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
913FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(I(2,0)))
ψ(M2(M)(M2+χ(M2)))
{n,n(1(1,,1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2)2,,1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2)2}
914FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(I(2,0)+1))
ψ(M2(M)(M2+χ(M2)+1))
{n,n(1,,1(1,,1,,2)2(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2)2}
915FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψΩI+1(I(2,0)+ψΩI+1(I(2,0)+I)))
ψ(M2χ(M+1)(M2+χ(M2)+ψ(M+1)(M2+χ(M2)+χ(M))))
{n,n(1(1,,1,,2)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
916FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψΩI2+1(I(2,0)+ψΩI2+1(I(2,0)+I2)))
ψ(M2χ(M×2+1)(M2+χ(M2)+ψχ(M×2+1)(M2+χ(M2)+χ(M×2))))
{n,n(1(1,,1,,3)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
917FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψΩI3+1(I(2,0)+ψΩI3+1(I(2,0)+I3)))
ψ(M2χ(M×3+1)(M2+χ(M2)+ψχ(M×3+1)(M2+χ(M2)+χ(M×3))))
{n,n(1(1,,1,,3)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
918FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+Iω)
ψ(M2+χ(M2)+χ(M×3))
{n,n(1(1,,1,,1,2)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
919FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(2,0)(0))
ψ(M2+χ(M2)+ψχ(M2)(M2))
{n,n(1(1,,1,,1(1,,1,,1,,2)2)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
920FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(2,0)(I(2,0)))
ψ(M2+χ(M2)+ψχ(M2)(M2+χ(M2)))
{n,n(1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
921FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ΩψI(2,0)(I(2,0))+1)
ψ(M2+χ(M2)+Ωψχ(M2)(M2+χ(M2))+1)
{n,n(1,,2,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
922FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(1,ψI(2,0)(I(2,0))+1)(0)) = ψ(I(2,0)+ΩΩΩ...ψI(2,0)(I(2,0))+1)
ψ(M2+χ(M2)+ψχ(M×(ψχ(M2)(M2+χ(M2))+1))(M×(ψχ(M2)(M2+χ(M2))+1))
{n,n(1,,1,,2(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
923FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+I(1,ψI(2,0)(I(2,0))×2))
ψ(M2+χ(M2)+χ(M×(ψχ(M2)(M2+χ(M2))×2))
{n,n(1,,1,,1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2(1,,1,,1,,2)1(1,,1,,1,,2)2)2}
924FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(2,0)(I(2,0)+1)) = ψ(I(2,0)+I(1,I(1,I(1,...ψI(2,0)(I(2,0))+1...))))
ψ(M2+χ(M2)+ψχ(M2)(M2+χ(M2)+1)) = ψ(M2+χ(M2)+χ(M×χ(M×χ(M×...(ψχ(M2)(M2+χ(M2))+1)...))))
{n,n(1,,1,,1(1,,1,,1,,2)2(1,,1,,1,,2)2)2}
925FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)+ψI(2,0)(I(2,0)+ψI(2,0)(I(2,0))))
ψ(M2+χ(M2)+ψχ(M2)(M2+χ(M2)+ψχ(M2)(M2+χ(M2))))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2(1,,1,,1,,2)2)2}
926FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0)×2) = ψ(I(2,0)+ψI(2,0)(I(2,0)+ψI(2,0)(I(2,0)+...)))
ψ(M2+χ(M2)×2) = ψ(M2+χ(M2)+χ(M2)) = ψ(M2+χ(M2)+ψχ(M2)(M2+χ(M2)+ψχ(M2)(M2+χ(M2)+ψχ(M2)(M2+χ(M2)+...))))
{n,n(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)3)2}
927FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI(2,0)+1) = ψ(I(0,I(2,0)+1)) = ψ(I(2,0)I(2,0)I(2,0)I(2,0)...) = ψ(εI(2,0)+1)
ψ(1st uncountable after 1st 2-weakly inaccessible)
ψ(M2+χ(M2+1)) = ψ(M2+χ(M2)χ(M2)χ(M2)...)
{n,n(1,,2,,1,,2)2}
928FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(2,0)+ψI(2,0)(0))
ψ(M2+χ(M2χ(M2)(M2)))
{n,n(1,,1(1,,1,,1(1,,1,,1,,2)2)2,,1,,2)2}
929FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(2,0)+ψI(2,0)(I(2,0)))
ψ(M2+χ(M2χ(M2)(M2+χ(M2))))
{n,n(1,,1(1,,1,,1(1,,1,,1,,2)1(1,,1,,1,,2)2)2,,1,,2)2}
930FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(2,0)×2) = ψ(ΩI(2,0)+I(2,0)) = ψ(ΩI(2,0)+ψI(2,0)I(2,0)+ψI(2,0)I(2,0)+...)))
ψ(M2+χ(M2+χ(M2)))
{n,n(1,,1(1,,1,,1,,2)2,,1,,2)2}
931FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,I(2,0)+1)(0)) = ψ(ψII(2,0)+1(0)) = ψ(ΩΩΩ...I(2,0)+1) = ψ(I(0,I(0,I(0,I(0,...I(2,0)+1))))
ψ(1st fixed point of uncountable after 1st 2-weakly inaccessible)
ψ(M2+M) = ψ(M2+χ(M2+χ(M2+χ(M2+χ(M2+...)))))
{n,n(1,,1,,2,,2)2}
932FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,I(2,0)+1)) = ψ(II(2,0)+1)
ψ(1st weakly inaccessible after 1st 2-weakly inaccessible)
ψ(M2+M+χ(M2+M))
{n,n(1,,1,,1(1,,1,,2)2,,2)2}
933FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(2,0)×2)) = ψ(II(2,0)×2)
ψ(M2+M×2+χ(M2+M×2))
{n,n(1,,1,,1(1,,1,,1,,2)2,,2)2}
934FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,1)(0)) = ψ(I(1,I(1,I(1,I(1,...I(2,0)+1...))))) = ψ(III...I(2,0)+1)
ψ(1st fixed point of weakly inaccessible after 1st 2-weakly inaccessible)
ψ(M2×2) = ψ(M2+M×M) = ψ(M2+M×χ(M×χ(M×χ(M×...))))
{n,n(1,,1,,1,,3)2}
935FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,1)) = ψ(ψI(2,1)(I(2,1))) = ψ(ψI(2,1)I(2,1)I(2,1)I(2,1)(...)))))
ψ(2nd 2-weakly inaccessible)
ψ(M2×2+χ(M2×2))
{n,n(1,,1,,1(1,,1,,1,,3)1(1,,1,,1,,3)2,,2)2}
936FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,2)(0)) = ψ(I(1,I(1,I(1,I(1,...I(2,1)+1...)))))
ψ(1st fixed point of weakly inaccessible after 2nd 2-weakly inaccessible)
ψ(M2×3)
{n,n(1,,1,,1,,4)2}
937FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,2)) = ψ(ψI(2,2)(I(2,2))) = ψ(ψI(2,2)I(2,2)I(2,2)I(2,2)(...)))))
ψ(3d 2-weakly inaccessible)
ψ(M2×3+χ(M2×3))
{n,n(1,,1,,1(1,,1,,1,,4)1(1,,1,,1,,4)2,,2)2}
938FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,ω))
ψ(ω-th 2-weakly inaccessible)
ψ(M2×ω)
{n,n(1,,1,,1,,1,2)2}
939FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,I(2,0)))
ψ(2-weakly inaccessible-th 2-weakly inaccessible)
ψ(M2×χ(M2))
{n,n(1,,1,,1,,1(1,,1,,1,,2)2)2}
940FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(3,0)(0)) = ψ(I(2,I(2,I(2,I(2,I(2,I(2,...)))))))
ψ(1st fixed point of 2-weakly inaccessible)
ψ(M3) = ψ(M2×M) = ψ(M2×χ(M2×χ(M2×χ(M2×...))))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)2)2} = {n,n(1,,1,,1,,1,,2)2}
941FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩψI(3,0)(0)+1)
ψ(1st uncountable after 1st fixed point of 2-weakly inaccessible)
ψ(M3ψχ(M3)(M3)+1)
{n,n(1,,2,,1,,1(1,,1,,1,,1,,2)2)2}
942FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(3,0)(0)+1)(0)) = ψ(ΩΩΩ...ψI(3,0)(0)+1)
ψ(1st fixed point of uncountable after 1st fixed point of 2-weakly inaccessible)
ψ(M3χ(M×(ψχ(M3)(M3)+1))(M×(ψχ(M3)(M3)+1)))
{n,n(1,,1,,2,,1(1,,1,,1,,1,,2)2)2}
943FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,ψI(3,0)(0)+1)(0)) = ψ(III...ψI(3,0)(0)+1)
ψ(1st fixed point of weakly inaccessible after 1st fixed point of 2-weakly inaccessible)
ψ(M3χ(M2×(ψχ(M3)(M3)+1))(M2×(ψχ(M3)(M3)+1)))
{n,n(1,,1,,1,,2(1,,1,,1,,1,,2)2)2}
944FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,ψI(3,0)(0)+1))
ψ(1st 2-weakly inaccessible after 1st fixed point of 2-weakly inaccessible)
ψ(M3+χ(M2×ψχ(M3)(M3)+1))
{n,n(1,,1,,1,,1(1,,1,,1,,2)2(1,,1,,1,,1,,2)2)2}
945FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,ψI(3,0)(0)×2))
ψ(M3+χ(M2×ψχ(M3)(M3)×2))
{n,n(1,,1,,1,,1(1,,1,,1,,2(1,,1,,1,,1,,2)2)2(1,,1,,1,,1,,2)2)2}
946FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(3,0)(1))
ψ(2nd fixed point of 2-weakly inaccessible)
ψ(M3χ(M3)(M3+1)) =ψ(M3+χ(M2×χ(M2×χ(M2×...(ψχ(M3)(M3)+1)...))))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)3)2}
947FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(3,0)I(3,0)(0))) = ψ(M3χ(M3)(M3))
ψ(1st fixed point of 2-weakly inaccessible-th fixed point of 2-weakly inaccessible)
ψ(M3χ(M3)(M3χ(M3)(M3)))) = ψ(M3χ(M3)(M3+χ(M2×χ(M2×χ(M2×...)))))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)1(1,,1,,1,,1(1,,1,,1,,1,,2)2)2)2}
948FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(3,0)) = ψ(ψI(3,0)(I(3,0))) = ψ(ψI(3,0)I(3,0)I(3,0)I(3,0)(...)))))
ψ(1st 3-weakly inaccessible)
ψ(M3+χ(M3)) = ψ(M3χ(M3)(M3χ(M3)(M3χ(M3)(M3χ(M3)(M3+...))))))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)1(1,,1,,1,,1,,2)2)2}
949FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ΩI(3,0)+1)
ψ(1st uncountable after 1st 3-weakly inaccessible)
ψ(M3+χ(M3+1))
{n,n(1,,2,,1,,1,,2)2}
950FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(3,0)×2)
ψ(M3+χ(M3+χ(M3)))
{n,n(1,,1(1,,1,,1,,1,,2)2,,1,,1,,2)2}
951FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,I(3,0)+1)(0)) = ψ(ΩΩΩ...I(3,0)+1)
ψ(1st fixed point of uncountable after 1st 3-weakly inaccessible)
ψ(M3+M) = ψ(M3+χ(M3+χ(M3+χ(M3+χ(M3+...)))))
{n,n(1,,1,,2,,1,,2)2}
952FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,I(3,0)+1)) = ψ(II(3,0)+1)
ψ(1st weakly inaccessible after 1st 3-weakly inaccessible)
ψ(M3+M+χ(M3+M))
{n,n(1,,1,,1(1,,1,,2)2,,1,,2)2}
953FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(3,0)×2)) = ψ(II(3,0)×2)
ψ(M3+M×2+χ(M3+M×2))
{n,n(1,,1,,1(1,,1,,1,,1,,2)2,,1,,2)2}
954FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,I(3,0)+1)(0)) = ψ(III...I(3,0)+1)
ψ(1st fixed point of weakly inaccessible after 1st 3-weakly inaccessible)
ψ(M3+M2) = ψ(M3+M×M) = ψ(M3+M×χ(M×χ(M×χ(M×...))))
{n,n(1,,1,,1,,2,,2)2}
955FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,I(3,0)+1))
ψ(1st 2-weakly inaccessible after 1st 3-weakly inaccessible)
ψ(M3+M2+χ(M3+M2))
{n,n(1,,1,,1,,1(1,,1,,1,,2)2,,2)2}
956FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,I(3,0)×2))
ψ(M3+M2×2+χ(M3+M2×2))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)2,,2)2}
957FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(3,1)(0)) = ψ(I(2,I(2,I(2,I(2,I(2,...I(3,0)+1...)))))))
ψ(M3×2) = ψ(M3+M3) = ψ(M2+M2×M) = ψ(M2+M2×χ(M2×χ(M2×χ(M2×...))))
{n,n(1,,1,,1,,1,,3)2}
958FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(3,1)) = ψ(ψI(3,1)I(3,1)I(3,1)(...))))
ψ(2nd 3-weakly inaccessible)
ψ(M3×2+χ(M3×2))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,3)1(1,,1,,1,,1,,3)2,,2)2}
959FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(3,I(3,0))) = ψ(I(3,I(3,I(3,I(3,I(3,I(3,...)))))))
ψ(3-weakly inaccessible-th 3-weakly inaccessible)
ψ(M3×χ(M3))
{n,n(1,,1,,1,,1,,1(1,,1,,1,,1,,2)2)2}
960FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(4,0)(0)) = ψ(I(3,I(3,I(3,I(3,I(3,I(3,...)))))))
ψ(1st fixed point of 3-weakly inaccessible)
ψ(M4) = ψ(M3×M) = ψ(M3×χ(M3×χ(M3×χ(M3×...))))
{n,n(1,,1,,1,,1,,1(1,,1,,1,,1,,1,,2)2)2} = {n,n(1,,1,,1,,1,,1,,2)2}
961FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(4,0)) = ψ(ψI(4,0)(I(4,0))) = ψ(ψI(4,0)I(4,0)I(4,0)I(4,0)(...)))))
ψ(4-weakly inaccessible)
ψ(M4+χ(M4))
{n,n(1,,1,,1,,1,,1(1,,1,,1,,1,,1,,2)1(1,,1,,1,,1,,1,,2)2)2}
962FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(5,0)(0)) = ψ(I(4,I(4,I(4,I(4,I(4,I(4,...)))))))
ψ(1st fixed point of 4-weakly inaccessible)
ψ(M5) = ψ(M4×M) = ψ(M4×χ(M4×χ(M4×χ(M4×...))))
{n,n(1,,1,,1,,1,,1,,1(1,,1,,1,,1,,1,,1,,2)2)2} = {n,n(1,,1,,1,,1,,1,,1,,2)2}
963FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(5,0)) = ψ(ψI(5,0)(I(5,0))) = ψ(ψI(5,0)I(5,0)I(5,0)I(5,0)(...)))))
ψ(5-weakly inaccessible)
ψ(M5+χ(M5))
{n,n(1,,1,,1,,1,,1,,1(1,,1,,1,,1,,1,,1,,2)1(1,,1,,1,,1,,1,,1,,2)2)2}
964FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)) = ψ(I(sup(n),0))
ψ(1st fixed point of n-weakly inaccessible)
ψ(Mω)
{n,n(1(2,,)2)2}
965FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+Ω) = ψ(ψI(ω,0)(0))ψ(ψI(ω,0)(0))ψ(ψI(ω,0)(0))...
ψ(Mω+Ω)
{n,n(1'2(1(2,,)2)2)2}
966FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ23))
ψ(MωΩ23))
{n,n(1(1,,3)2(1(2,,)2)2)2}
967FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(0)))
ψ(MωΩ2χ(M)(M)))
{n,n(1(1,,1,,2)2(1(2,,)2)2)2}
968FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(2,0)(0)))
ψ(MωΩ2χ(M2)(M)))
{n,n(1(1,,1,,1,,2)2(1(2,,)2)2)2}
969FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(ω,0)(0)))
ψ(MωΩ2χ(Mω)(M)))
{n,n(1(1(2,,)2)3)2}
970FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(ω,0)(0)+ψΩ2I(ω,0)(0))))
ψ(MωΩ2χ(Mω)(M)+ψΩ2χ(Mω)(M))))
{n,n(1(1(2,,)2)1(1(2,,)2)2)2}
971FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(ω,0)(0)+ψΩ2I(ω,0)(0)+1)))
ψ(MωΩ2χ(Mω)(M)+ψΩ2χ(Mω)(M)+1)))
{n,n(1(2(2,,)2)2)2}
972FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ2I(ω,0)(0)+ψΩ2I(ω,0)(0)+Ω)))
ψ(MωΩ2χ(Mω)(M)+ψΩ2χ(Mω)(M)+Ω)))
{n,n(1'2(2,,)2)2}
973FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩ3I(ω,0)(0)+ψΩ3I(ω,0)(0)+Ω2)))
ψ(MωΩ3χ(Mω)(M)+ψΩ3χ(Mω)(M)+Ω2)))
{n,n(1(1,,3)2(2,,)2)2}
974FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+Ωω)
ψ(Mωω))
{n,n(1(1,,1,2)2(2,,)2)2}
975FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψI(0))
ψ(Mωχ(M)(M))
{n,n(1(1,,1(1,,1,,2)2)2,,1(1(2,,)2)2)2}
976FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψII+1))
ψ(Mωχ(M)(M+χ(M+1)))
{n,n(1(1,,1(1,,2,,2)2)2,,1(1(2,,)2)2)2}
977FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψII2(0)))
ψ(Mωχ(M)(M×2))
{n,n(1(1,,1(1,,1,,3)2)2,,1(1(2,,)2)2)2}
978FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψII(2,0)(0)))
ψ(Mωχ(M)(M2))
{n,n(1(1,,1(1,,1,,1,,2)2)2,,1(1(2,,)2)2)2}
979FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψII(ω,0)(0)))
ψ(Mωχ(M)(Mω))
{n,n(1(1,,1(1(2,,)2)2)2,,1(1(2,,)2)2)2}
980FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ΩψII(ω,0)(0))+1)
ψ(Mωψχ(M)(Mω)+1)
{n,n(1,,2(1(2,,)2)2)2}
981FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψII(ω,0)(0)+1)) = ψ(ψI(ω,0)(0)+ΩΩΩ...ψII(ω,0)(0))+1)
ψ(Mωχ(M)(Mω+1))
{n,n(1,,1(1,,1,,2)2(1(2,,)2)2)2}
982FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+I) = ψ(ψI(ω,0)(0)+ψII(ω,0)(0)+ψII(ω,0)(0)+ψII(ω,0)(0)+...))))
ψ(Mω+χ(M))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1(2,,)2)2)2}
983FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI+1I+1)) = ψ(ψI(ω,0)(0)+IIII...)
ψ(Mω+χ(M+1))
{n,n(1,,1(1,,2,,2)2(1(2,,)2)2)2}
984FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI+1I(ω,0)(0)))
ψ(Mωχ(M+1)(Mω))
{n,n(1,,1(1(2,,)2)3)2}
985FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI+1I(ω,0)(0)+ψΩI+1I(ω,0)(0))))
ψ(Mωχ(M+1)(Mωχ(M+1)(Mω)))
{n,n(1,,1(1(2,,)2)1(1(2,,)2)2)2}
986FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI+1I(ω,0)(0)+ψΩI+1I(ω,0)(0)+I)))
ψ(Mωχ(M+1)(Mωχ(M+1)(Mω+χ(M))))
{n,n(1(1,,1,,2)2(2,,)2)2}
987FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI2+1I(ω,0)(0)+ψΩI2+1I(ω,0)(0)+I2)))
ψ(Mωχ(M×2+1)(Mωχ(M×2+1)(Mω+χ(M×2))))
{n,n(1(1,,1,,3)2(2,,)2)2}
988FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+Iω)
ψ(Mωχ(M×ω)(M×ω))
{n,n(1(1,,1,,1,2)2,,1,,1(1(2,,)2)2)2}
989FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψI(2,0)(0))
ψ(Mωχ(M2)(M2))
{n,n(1(1,,1,,1,,2)2,,1,,1(1(2,,)2)2)2}
990FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψI(2,0)I(ω,0)(0)))
ψ(Mωχ(M2)(Mω))
{n,n(1(1,,1,,1(1(2,,)2)2)2,,1,,1(1(2,,)2)2)2}
991FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ΩψI(2,0)I(ω,0)(0))+1)
ψ(Mωψχ(M2)(Mω)+1)
{n,n(1,,2,,1(1(2,,)2)2)2}
992FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψI(1,ψI(2,0)I(ω,0)(0))+1)(0)) = ψ(ψI(ω,0)(0)+ψIψI(2,0)I(ω,0)(0))+1(0)) = ψ(ψI(ω,0)(0)+ΩΩΩ...ψI(2,0)I(ω,0)(0))+1)
ψ(Mωχ(M×(ψχ(M2)(Mω)+1))(M×(ψχ(M2)(Mω)+1)))
{n,n(1,,1,,2(1(2,,)2)2)2}
993FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψI(2,0)I(ω,0)(0)+1)) = ψ(ψI(ω,0)(0)+III...ψI(2,0)I(ω,0)(0))+1)
ψ(Mωχ(M2)(Mω)+1)
{n,n(1,,1,,1(1,,1,,1,,2)2(1(2,,)2)2)2}
994FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI(2,0)+1I(ω,0)(0)))
ψ(Mωχ(M2+1)(Mω))
{n,n(1,,1,,1(1(2,,)2)3)2}
995FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI(2,0)+1I(ω,0)(0)+ψΩI(2,0)+1I(ω,0)(0)+I(2,0))))
ψ(Mωχ(M2+1)(Mωχ(M2+1)(Mω+χ(M2))))
{n,n(1(1,,1,,1,,2)2(2,,)2)2}
996FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)+ψΩI(3,0)+1I(ω,0)(0)+ψΩI(3,0)+1I(ω,0)(0)+I(3,0))))
ψ(Mωχ(M3+1)(Mωχ(M3+1)(Mω+χ(M3))))
{n,n(1(1,,1,,1,,1,,2)2(2,,)2)2}
997FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(0)×2) = ψ(ψI(ω,0)(0)+ψI(ω,0)(0))
ψ(Mωχ(Mω)(Mω))
{n,n(1(1(2,,)2)2(2,,)2)2}
998FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI(ω,0)(0)+1) = ψ(ψI(ω,0)(0)ψI(ω,0)(0)ψI(ω,0)(0)...)
ψ(Mωψχ(Mω)(Mω)+1)
{n,n(1,,2(2,,)2)2}
999FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI(ω,0)(0)×2)
ψ(Mωψχ(Mω)(Mω)×2)
{n,n(1,,1(1(2,,)2)2(2,,)2)2}
1000FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψIψI(ω,0)(0))+1(0)) = ψ(ψI(1,ψI(ω,0)(0)+1)(0)) = ψ(ΩΩΩ...ψI(ω,0)(0)+1)
ψ(Mωχ(M×(ψχ(Mω)(Mω)+1))(M×(ψχ(Mω)(Mω)+1)))
{n,n(1,,1,,2(2,,)2)2}
1001FGH ψ-function
FGH ψ(ψM)
SAN
ψ(IψI(ω,0)(0)+1) = ψ(I(1,ψI(ω,0)(0)+1)) = ψ(ψI(1,ψI(ω,0)(0)+1)I(1,ψI(ω,0)(0)+1)I(1,ψI(ω,0)(0)+1)(...))))
ψ(Mω+χ(M×ψχ(Mω)(Mω)+1))
{n,n(1,,1,,1(1,,1,,2)2(2,,)2)2}
1002FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,ψI(ω,0)(0)×2))
ψ(Mω+χ(M×ψχ(Mω)(Mω)×2))
{n,n(1,,1,,1(1(2,,)2)2(2,,)2)2}
1003FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,ψI(ω,0)(0)+1)(0)) = ψ(III...ψI(ω,0)(0)+1)
ψ(Mωχ(M2×(ψχ(Mω)(Mω)+1))(M2×(ψχ(Mω)(Mω)+1)))
{n,n(1,,1,,1,,2(2,,)2)2}
1004FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,ψI(ω,0)(0)+1)) = ψ(ψI(2,ψI(ω,0)(0)+1)I(2,ψI(ω,0)(0)+1)I(2,ψI(ω,0)(0)+1)(...))))
ψ(Mω+χ(M2×ψχ(Mω)(Mω)+1))
{n,n(1,,1,,1(1,,1,,1,,2)2(2,,)2)2}
1005FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,ψI(ω,0)(0)×2))
ψ(Mω+χ(M2×ψχ(Mω)(Mω)×2))
{n,n(1,,1,,1,,1(1(2,,)2)2(2,,)2)2}
1006FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(3,ψI(ω,0)(0)+1)(0)) = ψ(I(2,I(2,I(2,I(2,...)))))
ψ(Mωχ(M3×(ψχ(Mω)(Mω)+1))(M3×(ψχ(Mω)(Mω)+1)))
{n,n(1,,1,,1,,1,,2(2,,)2)2}
1007FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(3,ψI(ω,0)(0)+1)) = ψ(ψI(3,ψI(ω,0)(0)+1)I(3,ψI(ω,0)(0)+1)I(3,ψI(ω,0)(0)+1)(...))))
ψ(Mω+χ(M3×ψχ(Mω)(Mω)+1))
{n,n(1,,1,,1,,1(1,,1,,1,,1,,2)2(2,,)2)2}
1008FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(3,ψI(ω,0)(0)×2))
ψ(Mω+χ(M3×ψχ(Mω)(Mω)×2))
{n,n(1,,1,,1,,1,,1(1(2,,)2)2(2,,)2)2}
1009FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(1)) = ψ(I(sup(n),ψI(ω,0)(0))
ψ(Mωχ(Mω)(Mω+1))
{n,n(1(2,,)3)2}
1010FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(2)) = ψ(I(sup(n),ψI(ω,0)(1))
ψ(Mωχ(Mω)(Mω+2))
{n,n(1(2,,)4)2}
1011FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(ω))
ψ(Mωχ(Mω)(Mω+ω))
{n,n(1(2,,)1,2)2}
1012FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)(Ω)) = ψ(ψI(ω,0)(ψ(ψI(ω,0)(ψ(ψI(ω,0)(ψ(ψI(ω,0)(...))))))))
ψ(Mωχ(Mω)(Mω+Ω))
{n,n(1(2,,)1'2)2}
1013FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0)I(ω,0)(0)))
ψ(Mωχ(Mω)(Mωχ(Mω)(Mω)))
{n,n(1(2,,)1(1(2,,)2)2)2}
1014FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(ω,0)) = ψ(ψI(ω,0)I(ω,0)I(ω,0)I(ω,0)(...))))) = ψ(ψχ(Mω)(χ(Mω))) = ψ(χ(Mω)) = ψ(Mω)
ψ(ω-weakly inaccessible)
ψ(Mω+χ(Mω)) = ψ(Mωχ(Mω)(Mωχ(Mω)(Mωχ(Mω)(Mω+...))))
{n,n(1(2,,)1(1(2,,)1,,2)2)2} = {n,n(1(2,,)1,,2)2}
1015FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(ω,0)(0))
ψ(Mω+χ(Mω)+ψχ(Mω)(Mω))
{n,n(1(1(2,,)2)2(2,,)1(1(2,,)1,,2)2)2}
1016FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(ω,0)(I(ω,0)))
ψ(Mω+χ(Mω)+ψχ(Mω)(Mω+χ(Mω)))
{n,n(1(1(2,,)1(1(2,,)1,,2)2)2(2,,)1(1(2,,)1,,2)2)2}
1017FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ΩψI(ω,0)(I(ω,0))+1)
ψ(Mω+χ(Mω)+Ωψχ(Mω)(Mω+χ(Mω))+1)
{n,n(1,,2(2,,)1(1(2,,)1,,2)2)2}
1018FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(1,ψI(ω,0)(I(ω,0))+1)(0))
ψ(Mω+χ(Mω)+ψχ(M×(ψχ(Mω)(Mω+χ(Mω))+1))(M×(ψχ(Mω)(Mω+χ(Mω))+1))
{n,n(1,,1,,2(2,,)1(1(2,,)1,,2)2)2}
1019FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(2,ψI(ω,0)(I(ω,0))+1)(0))
ψ(Mω+χ(Mω)+ψχ(M2×(ψχ(Mω)(Mω+χ(Mω))+1))(M2×(ψχ(Mω)(Mω+χ(Mω))+1))
{n,n(1,,1,,1,,2(2,,)1(1(2,,)1,,2)2)2}
1020FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(3,ψI(ω,0)(I(ω,0))+1)(0))
ψ(Mω+χ(Mω)+ψχ(M3×(ψχ(Mω)(Mω+χ(Mω))+1))(M3×(ψχ(Mω)(Mω+χ(Mω))+1))
{n,n(1,,1,,1,,1,,2(2,,)1(1(2,,)1,,2)2)2}
1021FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)+ψI(ω,0)(I(ω,0)+1))
ψ(Mω+χ(Mω)+ψχ(Mω)(Mω+1))
{n,n(1(2,,)2(1(2,,)1,,2)2)2}
1022FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0)×2) = ψ(I(ω,0)+I(ω,0)) = ψ(I(ω,0)+ψI(ω,0)(I(ω,0)+ψI(ω,0)(I(ω,0)+ψI(ω,0)(I(ω,0)+...))))
ψ(Mω+χ(Mω)×2) = ψ(Mω+χ(Mω)+χ(Mω)) = ψ(Mω+χ(Mω)+ψχ(Mω)(Mωχ(Mω)(Mωχ(Mω)(Mω+...))))
{n,n(1(2,,)1(1(2,,)1,,2)3)2}
1023FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(ω,0)+1) = ψ(I(ω,0)I(ω,0)I(ω,0)I(ω,0)...)
ψ(Mω+χ(Mω+1)) = ψ(Mω+χ(Mω)χ(Mω)χ(Mω)...)
{n,n(1,,2(2,,)1,,2)2}
1024FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(ω,0)+1)(0)) = ψ(ΩΩΩ...I(ω,0)+1)
ψ(Mω+M) = ψ(Mω+χ(Mω+χ(Mω+χ(Mω+χ(Mω+...)))))
{n,n(1,,1,,2(2,,)1,,2)2}
1025FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(ω,0)+1) = ψ(ψI(1,I(ω,0)+1)I(1,I(ω,0)+1)I(1,I(ω,0)+1)(...))))
ψ(Mω+M+χ(Mω+M))
{n,n(1,,1,,1(1,,1,,2)2(2,,)1,,2)2}
1026FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,I(ω,0)+1)(0)) = ψ(III...I(ω,0)+1)
ψ(Mω+M2) = ψ(Mω+M×χ(M×χ(M×χ(M×...))))
{n,n(1,,1,,1,,2(2,,)1,,2)2}
1027FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,I(ω,0)+1) = ψ(ψI(2,I(ω,0)+1)I(2,I(ω,0)+1)I(2,I(ω,0)+1)(...))))
ψ(Mω+M2+χ(Mω+M2))
{n,n(1,,1,,1(1,,1,,1,,2)2(2,,)1,,2)2}
1028FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,1)(0)) = ψ(I(sup(n),I(ω,0)+1))
ψ(Mω×2) = ψ(Mω+Mω)
{n,n(1(2,,)2,,2)2}
1029FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,1)) = ψ(ψI(ω,1)I(ω,1)I(ω,1)(...))))
ψ(Mω×2+χ(Mω×2))
{n,n(1(2,,)1,,3)2}
1030FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,2)(0)) = ψ(I(sup(n),I(ω,1)+1))
ψ(Mω×3)
{n,n(1(2,,)2,,3)2}
1031FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,2)) = ψ(ψI(ω,2)I(ω,2)I(ω,2)(...))))
ψ(Mω×3+χ(Mω×3))
{n,n(1(2,,)1,,4)2}
1032FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,ψI(ω,0)(0)))
ψ(Mω×ψχ(Mω)(Mω))
{n,n(1(2,,)1,,1(1(2,,)2)2)2}
1033FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,ψI(ω,0)(I(ω,0))))
ψ(Mω×ψχ(Mω)(Mω+χ(Mω)))
{n,n(1(2,,)1,,1(1(2,,)1(1(2,,)1,,2)2)2)2}
1034FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,ψI(ω,0)(I(ω,ψI(ω,0)(0)))))
ψ(Mω×ψχ(Mω)(Mω+χ(Mω×ψχ(Mω)(Mω))))
{n,n(1(2,,)1,,1(1(2,,)1(1(2,,)1,,1(1(2,,)2)2)2)2)2}
1035FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,I(ω,0))) = ψ(I(ω,ψI(ω,0)(I(ω,ψI(ω,0)(I(ω,ψI(ω,0)(...)))))))
ψ(Mω×χ(Mω)) = ψ(Mω×ψχ(Mω)(Mω+χ(Mω×ψχ(Mω)(Mω×...))))
{n,n(1(2,,)1,,1(1(2,,)1,,2)2)2}
1036FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,I(ω,I(ω,0))))
ψ(Mω×χ(Mω×χ(Mω)))
{n,n(1(2,,)1,,1(1(2,,)1,,1(1(2,,)1,,2)2)2)2}
1037FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω+1,0)(0)) = ψ(I(ω,I(ω,I(ω,I(ω,I(ω,...))))))
ψ(Mω+1) = ψ(Mω×M) = ψ(Mω×χ(Mω×χ(Mω×χ(Mω×χ(Mω×...)))))
{n,n(1(2,,)1,,1,,2)2}
1038FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω+1,0)) = ψ(ψI(ω+1,0)I(ω+1,0)I(ω+1,0)(...))))
ψ(Mω+1+χ(Mω+1))
{n,n(1(2,,)1,,1,,1(1,,1,,2)2)2}
1039FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω+2,0)(0)) = ψ(I(ω+1,I(ω+1,I(ω+1,I(ω+1,I(ω+1,...))))))
ψ(Mω+2) = ψ(Mω+1×M)
{n,n(1(2,,)1,,1,,1,,2)2}
1040FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω+2,0)) = ψ(ψI(ω+2,0)I(ω+2,0)I(ω+2,0)(...))))
ψ(Mω+2+χ(Mω+2))
{n,n(1(2,,)1,,1,,1,,1(1,,1,,1,,2)2)2}
1041FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω×2,0)(0)) = ψ(I(...I(ω+4,I(ω+3,I(ω+2,I(ω+1,I(ω,0)))))...))
ψ(Mω×2)
{n,n(1(2,,)1(2,,)2)2}
1042FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω×2,0)) = ψ(ψI(ω×2,0)I(ω×2,0)I(ω×2,0)(...))))
ψ(Mω×2+χ(Mω×2))
{n,n(1(2,,)1(2,,)1,,2)2}
1043FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω2,0)(0)) = ψ(I(...I(ω×5,I(ω×4,I(ω×3,I(ω×2,I(ω,0)))))...))
ψ(Mω2)
{n,n(1(3,,)2)2}
1044FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω2,0)) = ψ(ψI(ω2,0)I(ω2,0)I(ω2,0)(...))))
ψ(Mω2+χ(Mω2))
{n,n(1(3,,)1,,2)2}
1045FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ε0,0)(0)) = ψ(ψI(ψ(Ω),0)(0))
ψ(Mε0)
{n,n(1(1(1'2)2,,)2)2}
1046FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ε0,0)) = ψ(I(ψ(Ω),0)) = ψ(ψI(ε0,0)I(ε0,0)I(ε0,0)(...))))
ψ(Mε0+χ(Mε0))
{n,n(1(1(1'2)2,,)1,,2)2}
1047FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(Ω,0)(0)) = ψ(ψI(ψ(ψI(ψ(ψI(...,0)(0)),0)(0)),0)(0)) = ψ(ψχ(MΩ)(0))
ψ(MΩ)
{n,n(1(1'2,,)2)2}
1048FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(Ω,0)) = ψ(ψI(Ω,0)I(Ω,0)I(Ω,0)(...))))
ψ(1st uncountable-weakly inaccessible)
ψ(MΩ+χ(MΩ))
{n,n(1(1'2,,)1,,2)2}
1049FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(Ω2,0)(0))
ψ(MΩ2)
{n,n(1(1(1,,3)2,,)2)2}
1050FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(Ω2,0)) = ψ(ψI(Ω2,0)I(Ω2,0)I(Ω2,0)(...))))
ψ(2nd uncountable-weakly inaccessible)
ψ(MΩ2+χ(MΩ2))
{n,n(1(1(1,,3)2,,)1,,2)2}
1051FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I,0)(0)) = ψ(ψI(ψII(ψII(...,0)(0)),0)(0)),0)(0))
ψ(Mχ(M))
{n,n(1(1(1,,1,,2)2,,)2)2}
1052FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(I,0)) = ψ(ψI(I,0)I(I,0)I(I,0)(0))))
ψ(1st weakly inaccessible-weakly inaccessible)
ψ(Mχ(M)+χ(Mχ(M)))
{n,n(1(1(1,,1,,2)2,,)1,,2)2}
1053FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(2,0),0)(0))
ψ(Mχ(M2))
{n,n(1(1(1,,1,,1,,2)2,,)2)2}
1054FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(2,0),0)) = ψ(ψI(I(2,0),0)I(I(2,0),0)I(I(2,0),0)(...))))
ψ(Mχ(M2)+χ(Mχ(M2)))
{n,n(1(1(1,,1,,1,,2)2,,)1,,2)2}
1055FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(ω,0)(0),0)(0)) = ψ(I(I(sup(n),0)))
ψ(Mχ(Mω))
{n,n(1(1(1(2,,)2)2,,)2)2}
1056FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(ω,0),0)) = ψ(ψI(ψI(ω,0)(0),0)I(ψI(ω,0)(0),0)I(ψI(ω,0)(0),0)(...))))
ψ(Mχ(Mω)+χ(Mχ(Mω)))
{n,n(1(1(1(2,,)1,,2)2,,)1,,2)2}
1057FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(ψI(ω,0)(0),0)(0),0)(0)) = ψ(I(I(I(sup(n),0))))
ψ(Mχ(Mχ(Mω)))
{n,n(1(1(1(1(1(2,,)2)2,,)2)2,,)2)2}
1058FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(I(ω,0),0),0)) = ψ(ψI(ψI(ψI(ω,0)(0),0)(0),0)I(ψI(ψI(ω,0)(0),0)(0),0)I(ψI(ψI(ω,0)(0),0)(0),0)(...))))
ψ(Mχ(Mχ(Mω))+χ(Mχ(Mχ(Mω))))
{n,n(1(1(1(1(1(2,,)1,,2)2,,)1,,2)2,,)1,,2)2}
1059FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)) = ψ(I(I(I(I(...,0),0),0),0))
ψ(1st fixed point α-weakly inaccessible)
ψ(MM) = ψ(Mχ(Mχ(Mχ(M...))))
{n,n(1(1(1(1,,2,,)2)2,,)2)2} = {n,n(1(1,,2,,)2)2}
1060FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)+ψI(ω,0)(0))
ψ(MMχ(Mω)(Mω))
{n,n(1(1(2,,)2)2(1(1(1,,2,,)2)2,,)2)2}
1061FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)+ψI(ω,0)I(1,0,0)(0)))
ψ(MMχ(Mω)(MM))
{n,n(1(1(2,,)1(1(1(1(1,,2,,)2)2,,)2)2)2(1(1(1,,2,,)2)2,,)2)2}
1062FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)+I(ω,0))
ψ(MM+χ(Mω))
{n,n(1(1(2,,)1,,2)2(1(1(1,,2,,)2)2,,)2)2}
1063FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)+ψI(ψI(ω,0)(0),0)(0))
ψ(MMχ(Mχ(Mω)(Mχ(Mω)))
{n,n(1(1(1(1(2,,)2)2,,)2)2(1(1(1,,2,,)2)2,,)2)2}
1064FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)+I(I(ω,0),0))
ψ(MM+χ(Mχ(Mω)))
{n,n(1(1(1(1(2,,)1,,2)2,,)1,,2)2(1(1(1,,2,,)2)2,,)2)2}
1065FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(0)×2)
ψ(MMχ(MM)(MM))
{n,n(1(1(1(1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)2)2,,)2)2}
1066FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩψI(1,0,0)(0)+1)
ψ(MMψχ(MM)(MM)+1)
{n,n(1,,2(1(1(1,,2,,)2)2,,)2)2}
1067FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(1,0,0)(0)+1)(0))
ψ(MMχ(M×(ψχ(MM)(MM)+1))(M×(ψχ(MM)(MM)+1)))
{n,n(1,,1,,2(1(1(1,,2,,)2)2,,)2)2}
1068FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,ψI(1,0,0)(0)+1)(0))
ψ(MMχ(Mω×(ψχ(MM)(MM)+1))(Mω×(ψχ(MM)(MM)+1)))
{n,n(1(2,,)2(1(1(1,,2,,)2)2,,)2)2}
1069FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,ψI(1,0,0)(0)+1))
ψ(MM+χ(Mω×ψχ(MM)(MM)+1)
{n,n(1(2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1070FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ψI(ω,0)(0),ψI(1,0,0)(0)+1))
ψ(MM+χ(Mψχ(Mω)(Mω)×ψχ(MM)(MM)+1)
{n,n(1(1(1(2,,)2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1071FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(ω,0),ψI(1,0,0)(0)+1))
ψ(MM+χ(Mχ(Mω)×ψχ(MM)(MM)+1)
{n,n(1(1(1(2,,)1,,2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1072FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ψI(ψI(ω,0)(0),0)(0),ψI(1,0,0)(0)+1))
ψ(MM+χ(Mχ(Mψχ(Mω)(Mω))×ψχ(MM)(MM)+1)
{n,n(1(1(1(1(1(2,,)2)2,,)2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1073FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0),1)(0)) = ψ(I(ψI(ψI(...,0)(0),0)(0),ψI(1,0,0)(0)+1))
ψ(MMχ(Mψχ(MM)(MM))(Mψχ(MM)(MM)))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1074FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0),1)(1)) = ψ(I(ψI(ψI(....ψI(1,0,0)(0)+1...,0)(0),0)(0),ψI(1,0,0)(0)+1)
ψ(MMχ(Mψχ(MM)(MM))(Mψχ(MM)(MM)+1))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)3(1(1(1,,2,,)2)2,,)2)2}
1075FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ψI(1,0,0)(0),1)) = ψ(ψI(ψI(1,0,0)(0),1)I(ψI(1,0,0)(0),1)I(ψI(1,0,0)(0),1)(...))))
ψ(MM+χ(Mψχ(MM)(MM)))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1076FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,I(ψI(1,0,0)(0),1)+1))
ψ(MM+χ(Mψχ(MM)(MM)+Mω))
{n,n(1(2,,)1,,2(1(1(1(1(1,,2,,)2)2,,)2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1077FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0),2)(0)) = ψ(I(ψI(ψI(...,0)(0),0)(0),I(ψI(1,0,0)(0),1)+1))
ψ(MMχ(Mψχ(MM)(MM)×2)(Mψχ(MM)(MM)×2))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)2,,2(1(1(1,,2,,)2)2,,)2)2}
1078FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0),2)(1)) = ψ(I(ψI(ψI(....ψI(1,0,0)(0)+1...,0)(0),0)(0),I(ψI(1,0,0)(0),1)+1)
ψ(MMχ(Mψχ(MM)(MM)×2)(Mψχ(MM)(MM)×2+1))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)5,,2(1(1(1,,2,,)2)2,,)2)2}
1079FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ψI(1,0,0)(0),2)) = ψ(ψI(ψI(1,0,0)(0),2)I(ψI(1,0,0)(0),2)I(ψI(1,0,0)(0),2)(...))))
ψ(MM+χ(Mψχ(MM)(MM)×2))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)1,,3(1(1(1,,2,,)2)2,,)2)2}
1080FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0)+1,0)(0)) = ψ(I(ψI(1,0,0)(0),I(ψI(1,0,0)(0),I(ψI(1,0,0)(0),...))))
ψ(MMχ(Mψχ(MM)(MM)+1)(Mψχ(MM)(MM)+1))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)1,,1,,2(1(1(1,,2,,)2)2,,)2)2}
1081FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0)×ω,0)(0))
ψ(MMχ(Mψχ(MM)(MM)×ω)(Mψχ(MM)(MM)×ω))
{n,n(1(2(1(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1082FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0)2,0)(0))
ψ(MMχ(Mψχ(MM)(MM)2)(Mψχ(MM)(MM)2))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)3,,)2(1(1(1,,2,,)2)2,,)2)2}
1083FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ΩψI(1,0,0)(0)+1,0)(0))
ψ(MMχ(MΩψχ(MM)(MM)+1)(MΩψχ(MM)(MM)+1))
{n,n(1(1(1,,2(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1084FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(ω,ψI(1,0,0)(0)+1),0)(0))
ψ(MMχ(Mχ(Mω×ψχ(MM)(MM)))(Mχ(Mω×ψχ(MM)(MM))))
{n,n(1(1(1(2,,)1,,2(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1085FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(ψI(1,0,0)(0),1)(0),0)(0)) = ψ(I(I(ψI(ψI(...,0)(0),0)(0),ψI(1,0,0)(0)+1)))
ψ(MMχ(Mχ(Mψχ(MM)(MM)))(Mχ(Mψχ(MM)(MM))))
{n,n(1(1(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1086FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(ψI(1,0,0)(0),1),0))
ψ(MM+χ(Mχ(Mψχ(MM)(MM))))
{n,n(1(1(1(1(1(1(1(1,,2,,)2)2,,)2)2,,)2(1(1(1,,2,,)2)2,,)2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1087FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(1)) = ψ(I(I(I(I(...I(ψI(1,0,0)(0),1)...,0),0),0),0))
ψ(MMχ(MM)(MM+1)) = ψ(MM+χ(Mχ(Mχ(Mχ(M...ψχ(MM)(MM))))))
{n,n(1(1(1(1,,2,,)2)2,,)3)2}
1088FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(2)) = ψ(I(I(I(I(...I(ψI(1,0,0)(1),1)...,0),0),0),0))
ψ(MMχ(MM)(MM+2))
{n,n(1(1(1(1,,2,,)2)2,,)4)2}
1089FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)(ω))
ψ(MMχ(MM)(MM+ω))
{n,n(1(1(1(1,,2,,)2)2,,)1,2)2}
1090FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0)I(1,0,0)(0)))
ψ(MMχ(MM)(MMχ(MM)(MM)))
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1(1,,2,,)2)2,,)2)2)2}
1091FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,0,0)) = ψ(ψI(1,0,0)I(1,0,0)I(1,0,0)I(1,0,0)(...)))))
ψ(hyper-weakly inaccessible or 1,0-hyper-weakly inaccessible)
ψ(MM+χ(MM)) = ψ(MMχ(MM)(MMχ(MM)(MMχ(MM)(MM+...))))
{n,n(1(1(1(1,,2,,)2)2,,)1,,2)2}
1092FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(1,0,0)+1)
ψ(MM+χ(MM+1))
{n,n(1,,2(1(1(1,,2,,)2)2,,)1,,2)2}
1093FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,I(1,0,0)+1)(0))
ψ(MM+Mω)
{n,n(1(2,,)2(1(1(1,,2,,)2)2,,)1,,2)2}
1094FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(1,0,0),1)(0))
ψ(MM+Mψχ(MM)(MM)+1)
{n,n(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)2(1(1(1,,2,,)2)2,,)1,,2)2}
1095FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(1,0,0),1)) = ψ(ψI(I(1,0,0),1)I(I(1,0,0),1)I(I(1,0,0),1)(...))))
ψ(MM+Mψχ(MM)(MM)+χ(MM+Mψχ(MM)(MM)))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1096FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,I(I(1,0,0),1)+1))
ψ(MM+Mψχ(MM)(MM)+Mω)
{n,n(1(2,,)1,,2(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)1,,2(1(1(1,,2,,)2)2,,)2)2}
1097FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(1,0,0),2)(0)) = ψ(I(ψI(ψI(...,0)(0),0)(0),I(1,0,0),1)+1))
ψ(MM+Mψχ(MM)(MM)×2)
{n,n(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)2,,2(1(1(1,,2,,)2)2,,)2)2}
1098FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(1,0,0)+1,0)(0)) = ψ(I(I(1,0,0),I(I(1,0,0),I(I(1,0,0),...))))
ψ(MM+Mψχ(MM)(MM)+1)
{n,n(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)1,,1,,2(1(1(1,,2,,)2)2,,)2)2}
1099FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(I(1,0,0),1)(0),0)(0)) = ψ(I(I(ψI(ψI(...,0)(0),0)(0),I(1,0,0)+1)))
ψ(MM+Mχ(Mψχ(MM)(MM)))
{n,n(1(1(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)2(1(1(1,,2,,)2)2,,)1,,2)2,,)2(1(1(1,,2,,)2)2,,)2)2}
1100FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,1)(0)) = ψ(I(I(I(I(...I(ψI(ψI(...,0)(0),0)(0),I(1,0,0)+1)...,0),0),0),0))
ψ(MM×2) = ψ(MM+MM)
{n,n(1(1(1(1,,2,,)2)2,,)2,,2)2}
1101FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,0,1)) = ψ(ψI(1,0,1)I(1,0,1)I(1,0,1)(...))))
ψ(2nd hyper-weakly inaccessible or 1,0-hyper-weakly inaccessible)
ψ(MM×2+χ(MM×2))
{n,n(1(1(1(1,,2,,)2)2,,)1,,3)2}
1102FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,1,0)(0)) = ψ(I(1,0,I(1,0,I(1,0,I(1,0,...)))))
ψ(1st fixed point hyper-weakly inaccessible or 1,0-hyper-weakly inaccessible)
ψ(MM+1) = ψ(MM×M)
{n,n(1(1(1(1,,2,,)2)2,,)1,,1,,2)2}
1103FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,1,0)) = ψ(ψI(1,1,0)I(1,1,0)I(1,1,0)(...))))
ψ(2-hyper-weakly inaccessible or 1,1-hyper-weakly inaccessible)
ψ(MM+1+χ(MM+1))
{n,n(1(1(1(1,,2,,)2)2,,)1,,1,,1(1,,1,,2)2)2}
1104FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,2,0)(0)) = ψ(I(1,1,I(1,1,I(1,1,I(1,1,...)))))
ψ(1st fixed point 2-hyper-weakly inaccessible or 1,1-hyper-weakly inaccessible)
ψ(MM+2)
{n,n(1(1(1(1,,2,,)2)2,,)1,,1,,1,,2)2}
1105FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,2,0)) = ψ(ψI(1,1,0)I(1,1,0)I(1,1,0)(...))))
ψ(3-hyper-weakly inaccessible or 1,2-hyper-weakly inaccessible)
ψ(MM+2+χ(MM+2)
{n,n(1(1(1(1,,2,,)2)2,,)1,,1,,1,,1(1,,1,,1,,2)2)2}
1106FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,ω,0)(0))
ψ(MM+ω)
{n,n(1(1(1(1,,2,,)2)2,,)1(2,,)2)2}
1107FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,ψI(1,0,0)(0),0)(0))
ψ(MM+ψχ(MM)(MM))
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1(1(1,,2,,)2)2,,)2)2,,)2)2}
1108FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(1,0,0),0))
ψ(MM+χ(MM))
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1(1(1,,2,,)2)2,,)1,,2)2,,)1,,2)2}
1109FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(2,0,0)(0)) = ψ(I(1,I(1,I(1,I(1,...,0),0),0),0))
ψ(1st fixed point α-hyper-weakly inaccessible or 1,α-hyper-weakly inaccessible)
ψ(MM×2) = ψ(MM+M)
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)2)2}
1110FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(2,0,0)) = ψ(ψI(2,0,0)I(2,0,0)I(2,0,0)(...))))
ψ(hyper-hyper-weakly inaccessible or 2,0-hyper-weakly inaccessible)
ψ(MM×2+χ(MM×2))
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)1,,2)2}
1111FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(3,0,0)(0)) = ψ(I(2,I(2,I(2,I(2,...,0),0),0),0))
ψ(1st fixed point α-hyper-hyper-weakly inaccessible or 2,α-hyper-weakly inaccessible)
ψ(MM×3)
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)2)2}
1112FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(3,0,0)) = ψ(ψI(3,0,0)I(3,0,0)I(3,0,0)(...))))
ψ(hyper-hyper-hyper-weakly inaccessible or 3,0-hyper-weakly inaccessible)
ψ(MM×3+χ(MM×3))
{n,n(1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)1(1(1(1,,2,,)2)2,,)1,,2)2}
1113FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0,0)(0)) = ψ(I(sup(n),0,0))
ψ(MM×ω)
{n,n(1(2(1(1,,2,,)2)2,,)2)2}
1114FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(ω,0,0)) = ψ(ψI(ω,0,0)I(ω,0,0)I(ω,0,0)(...))))
ψ(hyperω-weakly inaccessible or ω,0-hyper-weakly inaccessible)
ψ(MM×ω+χ(MM×ω))
{n,n(1(2(1(1,,2,,)2)2,,)1,,2)2}
1115FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0)(0),0,0)(0))
ψ(MM×ψχ(MM)(MM))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)2)2(1(1,,2,,)2)2,,)2)2}
1116FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(1,0,0),0,0)) = ψ(ψI(ψI(1,0,0)(0),0,0)I(ψI(1,0,0)(0),0,0)I(ψI(1,0,0)(0),0,0)(...))))
ψ(MM×χ(MM))
{n,n(1(1(1(1(1(1,,2,,)2)2,,)1,,2)2(1(1,,2,,)2)2,,)1,,2)2}
1117FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0)(0)) = ψ(I(I(I(I(...,0,0),0,0),0,0),0,0))
ψ(1st fixed point α,0-hyper-weakly inaccessible)
ψ(MM2) = ψ(MM×M)
{n,n(1(1(1(1,,2,,)2)3,,)2)2}
1118FGH ψ-function
ψ(cardinal)
FGH ψ(ψM)
SAN
ψ(I(1,0,0,0)) = ψ(ψI(1,0,0,0)I(1,0,0,0)I(1,0,0,0)(...))))
ψ(1,0,0-hyper-weakly inaccessible)
ψ(MM2+χ(MM2))
{n,n(1(1(1(1,,2,,)2)3,,)1,,2)2}
1119FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,1)(0))
ψ(MM2×2)
{n,n(1(1(1(1,,2,,)2)3,,)3,,2)2}
1120FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,0,1))
ψ(MM2×2+χ(MM2×2))
{n,n(1(1(1(1,,2,,)2)3,,)1,,3)2}
1121FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,1,0)(0))
ψ(MM2×M) = ψ(MM2+1)
{n,n(1(1(1(1,,2,,)2)3,,)1,,1,,2)2}
1122FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,1,0))
ψ(MM2+1+χ(MM2+1))
{n,n(1(1(1(1,,2,,)2)3,,)1,,1(1,,1,,2)2)2}
1123FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,1,0,0)(0))
ψ(MM2+M)
{n,n(1(1(1(1,,2,,)2)3,,)1(1(1(1,,2,,)2)2,,)2)2}
1124FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,1,0,0))
ψ(MM2+M+χ(MM2+M))
{n,n(1(1(1(1,,2,,)2)3,,)1(1(1(1,,2,,)2)2,,)1,,2)2}
1125FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,0,0,0)(0))
ψ(MM2×2)
{n,n(1(1(1(1,,2,,)2)3,,)1(1(1(1,,2,,)2)3,,)2)2}
1126FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(2,0,0,0))
ψ(MM2×2+χ(MM2×2))
{n,n(1(1(1(1,,2,,)2)3,,)1(1(1(1,,2,,)2)3,,)1,,2)2}
1127FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,0,0,0)(0))
ψ(MM2×ω)
{n,n(1(2(1(1,,2,,)2)3,,)2)2}
1128FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,0,0,0))
ψ(MM2×ω+χ(MM2×ω))
{n,n(1(2(1(1,,2,,)2)3,,)1,,2)2}
1129FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ψI(1,0,0,0)(0),0,0,0)(0))
ψ(MM2×ψχ(MM2)(MM2))
{n,n(1(1(1(1(1(1,,2,,)2)3,,)2)2(1(1,,2,,)2)3,,)2)2}
1130FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(1,0,0,0),0,0,0))
ψ(MM2×χ(MM2))
{n,n(1(1(1(1(1(1,,2,,)2)3,,)1,,2)2(1(1,,2,,)2)3,,)1,,2)2}
1131FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0)(0))
ψ(MM3)
{n,n(1(1(1(1,,2,,)2)4,,)2)2}
1132FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,0,0,0))
ψ(MM3+χ(MM3))
{n,n(1(1(1(1,,2,,)2)4,,)1,,2)2}
1133FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,0)(0))
ψ(MM4)
{n,n(1(1(1(1,,2,,)2)5,,)2)2}
1134FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,0,0,0,0))
ψ(MM4+χ(MM4))
{n,n(1(1(1(1,,2,,)2)5,,)1,,2)2}
1135FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)) = ψ(ψI(1ω)(0))
ψ(MMω)
{n,n(1(1(1(1,,2,,)2)1,2,,)2)2}
1136FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+ψI(0))
ψ(MMωχ(M)(M))
{n,n(1(1,,1,,2)2(1(1(1,,2,,)2)1,2,,)2)2}
1137FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+ψII(1,0,0,0,0,...)(0)))
ψ(MMωχ(M)(MMω))
{n,n(1(1(1(1(1,,2,,)2)1,2,,)2)2(1,,1,,2)2(1(1(1,,2,,)2)1,2,,)2)2}
1138FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+I)
ψ(MMω+χ(M)) = ψ(MMωχ(M)(MMωχ(M)(MMωχ(M)(...))))
{n,n(1,,1(1,,1,,2)1(1,,1,,2)2(1(1(1,,2,,)2)1,2,,)2)2}
1139FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+ψI(1,0,0)(0))
ψ(MMωχ(MM)(MM))
{n,n(1(1(1(1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1140FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+ψI(1,0,0)I(1,0,0,0,0,...)(0)))
ψ(MMωχ(MM)(MMω))
{n,n(1(1(1(1(1,,2,,)2)1,2,,)2)2(1(1(1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1141FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+I(1,0,0))
ψ(MMω+χ(MM)) = ψ(MMωχ(MM)(MMωχ(MM)(MMωχ(MM)(...))))
{n,n(1(1(1(1(1,,2,,)2)2,,)1,,2)2(1(1(1,,2,,)2)1,2,,)2)2}
1142FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+ψI(1,0,0,0)(0))
ψ(MMωχ(MM2)(MM2))
{n,n(1(1(1(1(1,,2,,)2)3,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1143FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)+I(1,0,0,0))
ψ(MMω+χ(MM2))
{n,n(1(1(1(1(1,,2,,)2)3,,)1,,2)2(1(1(1,,2,,)2)1,2,,)2)2}
1144FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(0)×2) = ψ(ψI(1,0,0,0,0,...)(0)+ψI(1,0,0,0,0,...)(0))
ψ(MMωχ(MMω)(MMω))
{n,n(1(1(1(1(1,,2,,)2)1,2,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1145FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,0,0,...)(1)) = ψ(ψI(1,0,0,0,0,...ψI(1,0,0,0,0,...)(0)+1)(0))
ψ(MMωχ(MMω)(MMω+1))
{n,n(1(1(1(1(1,,2,,)2)1,2,,)3)2(1(1(1,,2,,)2)1,2,,)2)2}
1146FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,0,0,0,...)) = ψ(ψI(1,0,0,0,0,...)I(1,0,0,0,0,...)I(1,0,0,0,0,...)(...)))) = ψ(I(1ω))
ψ(MMω+χ(MMω))
{n,n(1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1147FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(1,0,0,0,0,...)+1)
ψ(MMω+χ(MMω+1)) = ψ(MMωχ(MMω)+1)
{n,n(1,,3(1(1(1,,2,,)2)1,2,,)2)2}
1148FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩΩI(1,0,0,0,0,...)+1)
ψ(MMω+χ(MMω+χ(MMω)))
{n,n(1,,1(1,,2(1(1(1,,2,,)2)1,2,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1149FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+M) = ψ(MMω+χ(MMω+χ(MMω+...)))
{n,n(1,,1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1150FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(1,0,0,0,0,...)+1)(1))
ψ(MMω+M+ψχ(MMω+M)(MMω+M+1))
{n,n(1,,1(1,,1,,2(1(1(1,,2,,)2)1,2,,)2)3(1(1(1,,2,,)2)1,2,,)2)2}
1151FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,I(1,0,0,0,0,0,...)+1)) = ψ(II(1,0,0,0,0,0,...)+1)
ψ(MMω+M+χ(MMω+M)) = ψ(MMω+M+ψχ(MMω+M)(MMω+M+ψχ(MMω+M)(MMω+M+...)))
{n,n(1,,1(1,,1,,2(1(1(1,,2,,)2)1,2,,)2)1(1,,1,,2(1(1(1,,2,,)2)1,2,,)2)2(1(1(1,,2,,)2)1,2,,)2)2}
1152FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩII(1,0,0,0,0,...)+1)
ψ(MMω+M+χ(MMω+M+1)) = ψ(MMω+M+Ωχ(MMω+M)+1)
{n,n(1,,2,,2(1(1(1,,2,,)2)1,2,,)2)2}
1153FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(1,0,0,0,0,...)+2)(0))
ψ(MMω+M×2) = ψ(MMω+M+M) = ψ(MMω+M+χ(MMω+M+χ(MMω+M+...)))
{n,n(1,,1,,3(1(1(1,,2,,)2)1,2,,)2)2}
1154FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(1,0,0,0,0,...)+ω)(0))
ψ(MMω+M×ω)
{n,n(1,,1,,1,2(1(1(1,,2,,)2)1,2,,)2)2}
1155FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+M2)
{n,n(1,,1,,1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1156FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+Mω)
{n,n(1(2,,)2(1(1(1,,2,,)2)1,2,,)2)2}
1157FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ω,I(1,0,0,0,0,0,...)+1))
ψ(MMω+Mω+χ(MMω+Mω))
{n,n(1,,2(2,,)2(1(1(1,,2,,)2)1,2,,)2)2}
1158FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(ω,I(1,0,0,0,0,...)+1))(0))
ψ(MMω+Mω+M) = ψ(MMω+Mω+χ(MMω+Mω+χ(MMω+Mω+...)))
{n,n(1,,1,,2(2,,)2(1(1(1,,2,,)2)1,2,,)2)2}
1159FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω,I(1,0,0,0,0,...)+2)(0))
ψ(MMω+Mω×2)
{n,n(1(2,,)3(1(1(1,,2,,)2)1,2,,)2)2}
1160FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ω+1,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+Mω+1)
{n,n(1(2,,)1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1161FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+MM)
{n,n(1(1(1(1,,2,,)2)2,,)2(1(1(1,,2,,)2)1,2,,)2)2}
1162FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,I(1,0,0,0,0,...)+1)(1))
ψ(MMω+MMχ(MMω+MM)(MMω+MM+1))
{n,n(1(1(1(1,,2,,)2)2,,)5(1(1(1,,2,,)2)1,2,,)2)2}
1163FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1,0,I(1,0,0,0,0,...)+1))
ψ(MMω+MM+χ(MMω+MM)) = ψ(MMω+MMχ(MMω+MM)(MMω+MMχ(MMω+MM)(MMω+MM+...)))
{n,n(1(1(1(1,,2,,)2)2,,)1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1164FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ΩI(1,0,I(1,0,0,0,0,...)+1)+1)
ψ(MMω+MM+χ(MMω+MM+1)) = ψ(MMωχ(MMω+MM)+1)
{n,n(1,,2(1(1(1,,2,,)2)2,,)1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1165FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,I(1,0,I(1,0,0,0,0,...)+1)+1)(0))
ψ(MMω+MM+M) = ψ(MMω+MM+χ(MMω+MM+χ(MMω+MM+...)))
{n,n(1,,1,,2(1(1(1,,2,,)2)2,,)1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1166FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,I(1,0,0,0,0,...)+2)(0))
ψ(MMω+MM×2)
{n,n(1(1(1(1,,2,,)2)2,,)2,,2(1(1(1,,2,,)2)1,2,,)2)2}
1167FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,1,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+MM+1)
{n,n(1(1(1(1,,2,,)2)2,,)1,,1,,2(1(1(1,,2,,)2)1,2,,)2)2}
1168FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1,0,0,I(1,0,0,0,0,...)+1)(0))
ψ(MMω+MM2)
{n,n(1(1(1(1,,2,,)2)3,,)2(1(1(1,,2,,)2)1,2,,)2)2}
1169FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1ω10)(0))
ψ(MMω×2) = ψ(MMω+MMω)
{n,n(1(1(1(1,,2,,)2)1,2,,)3)2}
1170FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1ω11)(0))
ψ(MMω×M) = ψ(MMω+1)
{n,n(1(1(1(1,,2,,)2)1,2,,)1,,2)2}
1171FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(2ω)(0))
ψ(MMω×2)
{n,n(1(1(1(1,,2,,)2)1,2,,)1(1(1(1,,2,,)2)1,2,,)2)2}
1172FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(ωω)(0))
ψ(MMω×ω)
{n,n(1(2(1(1,,2,,)2)1,2,,)2)2}
1173FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(I(1,0,0,0,0...)ω)(0))
ψ(MMω×χ(MMω))
{n,n(1(1(1(1(1(1,,2,,)2)1,2,,)2)2(1(1,,2,,)2)1,2,,)2)2}
1174FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1ω+1)(0))
ψ(MMω+1) = ψ(MMω×M)) = ψ(MMω×χ(MMω×...))
{n,n(1(1(1(1,,2,,)2)2,2,,)2)2}
1175FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1ω2)(0))
ψ(MMω2)
{n,n(1(1(1(1,,2,,)2)1,1,2,,)2)2}
1176FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1Ω)(0))
ψ(MMΩ) = ψ(MMψ(MM...))
{n,n(1(1(1(1,,2,,)2)1'2,,)2)2}
1177FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I)(0))
ψ(MMI) = ψ(MMχ(M)) = ψ(MMψχ(M)(MM...))
{n,n(1(1(1(1,,2,,)2)1(1,,1,,2)2,,)2)2}
1178FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(0))
ψ(MMM) = ψ(MMχ(MM...))
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1179FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(0)+ψI(1ω)(0))
ψ(MMMχ(MMω)(χ(MMω)))
{n,n(1(1(1(1(1,,2,,)2)1,2,,)2)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1180FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(0)+I(1ω))
ψ(MMM+χ(MMω))
{n,n(1(1,,2(1(1(1,,2,,)2)1,2,,)2)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1181FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(0)+I(1I(1ω)))
ψ(MMM+χ(MMχ(MMω)))
{n,n(1(1,,2(1(1(1,,2,,)2)1(1,,2(1(1(1,,2,,)2)1,2,,)2)2,,)2)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1182FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(0)×2)
ψ(MMMχ(MMM)(MMM))
{n,n(1(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1183FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(ψI(1I(1I(1I(1...))))(0),1))
ψ(MMM+χ(MMψχ(MMM)(MMM)))
{n,n(1,,2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1184FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(I(ψI(1I(1I(1I(1...))))(0),1),0))
ψ(MMM+χ(MMχ(MMψχ(MMM)(MMM))))
{n,n(1,,2(1(1(1,,2,,)2(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1185FGH ψ-function
FGH ψ(ψM)
SAN
ψ(ψI(1I(1I(1I(1...))))(1)) = ψ(I(I(I(...I(ψI(1I(1I(1I(1...))))(0),1)...,0),0),0))
ψ(MMMχ(MMM)(MMM+1)) = ψ(MMM+χ(MMχ(MM...ψχ(MMM)(MMM))))
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)3)2}
1186FGH ψ-function
FGH ψ(ψM)
SAN
ψ(I(1I(1I(1I(1...))))) = ψ(ψI(1I(1I(1I(1...))))I(1I(1I(1I(1...))))I(1I(1I(1I(1...))))(...))))
ψ(MMM+χ(MMM)) = ψ(MMMχ(MMM)(MMMχ(MMM)(MMM+...)))
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,2)2}
1187FGH ψ(ψM)
SAN
ψ(MMM+χ(MMM+1))
{n,n(1,,2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,2)2}
1188FGH ψ(ψM)
SAN
ψ(MMM+M) = ψ(MMM+χ(MMM+χ(MMM+...)))
{n,n(1,,1,,2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,2)2}
1189FGH ψ(ψM)
SAN
ψ(MMM+MM)
{n,n(1(1(1(1,,2,,)2)2,,)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,2)2}
1190FGH ψ(ψM)
SAN
ψ(MMM+MMω)
{n,n(1(1(1(1,,2,,)2)1,2,,)2(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,2)2}
1191FGH ψ(ψM)
SAN
ψ(MMM×2)
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)3,,2)2}
1192FGH ψ(ψM)
SAN
ψ(MMM+1)
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)1,,1,,2)2}
1193FGH ψ(ψM)
SAN
ψ(MMM×ω)
{n,n(1(2(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1194FGH ψ(ψM)
SAN
ψ(MMM+1)
{n,n(1(1(1(1,,2,,)2)2(1(1,,2,,)2)2,,)2)2}
1195FGH ψ(ψM)
SAN
ψ(MMM×2)
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)3,,)2)2}
1196FGH ψ(ψM)
SAN
ψ(MMM2)
{n,n(1(1(1(1,,2,,)2)1(1(1,,2,,)2)1(1(1,,2,,)2)2,,)2)2}
1197FGH ψ(ψM)
SAN
ψ(MMMω)
{n,n(1(1(2(1,,2,,)2)2,,)2)2}
1198FGH ψ(ψM)
SAN
ψ(MMMM)
{n,n(1(1(1,,2,,)2)2(1,,2,,)2)2}
1199FGH ψ(ψM)
SAN
ψ(MMMMM)
{n,n(1(1(1,,2,,)2)1(1(1,,2,,)2)2(1,,2,,)2)2}
1200FGH ψ(ψM)
SAN
ψ(MMMMMM)
{n,n(1(1(1(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1201name
proof theory
FGH ψ(ψM)

ψ(cardinal)
FGH ψ(ψM2)
SAN
Rathjen ordinal
KPM
ψ(MMMMM...) = ψ(εM+1) = ψ(ΩM+1)
ψ(1st uncountable after 1st weakly Mahlo)
ψ(χM2(1))
{n,n(1,,2(1,,2,,)2)2}
1202FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(MM))
{n,n(1(1(1(1(1,,2,,)2)2,,)1,,2)2(1(1,,2(1,,2,,)2)2,,)2)2}
1203FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(MMMM))
{n,n(1(1(1(1(1(1,,2,,)2)2(1,,2,,)2)2,,)1,,2)2(1(1,,2(1,,2,,)2)2,,)2)2}
1204FGH ψ(ψM)
SAN
ψ(ΩM+1χ(ΩM+1)M+1)) = ψ(ΩM+1+χ(MMMMM...)) = ψ(ΩM+1+χ(εM+1))
{n,n(1(1(1(1,,2(1,,2,,)2)2,,)2)2(1(1,,2(1,,2,,)2)2,,)2)2}
1205FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(εM+1ψχ(ΩM+1)M+1)))
{n,n(1(1(1(1,,2(1,,2,,)2)2,,)2)3(1(1,,2(1,,2,,)2)2,,)2)2}
1206FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(εM+1χ(εM+1ψχ(ΩM+1)M+1)))
{n,n(1(1(1(1,,2(1,,2,,)2)2,,)2)4(1(1,,2(1,,2,,)2)2,,)2)2}
1207FGH ψ(ψM)
SAN
ψ(ΩM+1χ(ΩM+1)M+1+1)) = ψ(ΩM+1+χ(εM+1χ(εM+1χ(εM+1χ(εM+1...ψχ(ΩM+1)M+1))))))
{n,n(1(1(1(1,,2(1,,2,,)2)2,,)3)1,2(1(1,,2(1,,2,,)2)2,,)3)2}
1208FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(ΩM+1)) = ψ(ΩM+1χ(ΩM+1)M+1χ(ΩM+1)M+1χ(ΩM+1)M+1+...))))
{n,n(1,,2(1(1,,2(1,,2,,)2)2,,)2)2}
1209FGH ψ(ψM)
SAN
ψ(ΩM+1+χ(ΩM+1+1)) = ψ(ΩM+1χ(ΩM+1)+1)
{n,n(1,,3(1(1,,2(1,,2,,)2)2,,)2)2}
1210FGH ψ(ψM)
SAN
ψ(ΩM+1+M) = ψ(ΩM+1+χ(ΩM+1+χ(ΩM+1+χ(ΩM+1+...))))
{n,n(1,,1,,2(1(1,,2(1,,2,,)2)2,,)2)2}
1211FGH ψ(ψM)
SAN
ψ(ΩM+1+MM)
{n,n(1(1(1(1,,2,,)2)2,,)2(1(1,,2(1,,2,,)2)2,,)2)2}
1212FGH ψ(ψM)
SAN
ψ(ΩM+1+MMMM)
{n,n(1(1(1(1(1,,2,,)2)2(1,,2,,)2)2,,)2(1(1,,2(1,,2,,)2)2,,)2)2}
1213FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1)) = ψ(ΩM+1+MMMMM...)
{n,n(1(1(1,,2(1,,2,,)2)2,,)3)2}
1214FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1+1)) = ψ(ΩM+1ΩM+1M+1)×ω)
{n,n(1(1(1,,2(1,,2,,)2)2,,)1,2)2}
1215FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1+Ω)) = ψ(ΩM+1ΩM+1M+1)×Ω)
{n,n(1(1(1,,2(1,,2,,)2)2,,)1'2)2}
1216FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1+M))=ψ(ΩM+1ΩM+1M+1)×M)
{n,n(1(1(1,,2(1,,2,,)2)2,,)1,,2)2}
1217FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1ΩM+1M+1))) = ψ(ΩM+1ΩM+1M+1)2)
{n,n(1(1(1,,2(1,,2,,)2)2,,)1(1(1,,2(1,,2,,)2)2,,)2)2}
1218FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1ΩM+1M+1+1))) = ψ(ΩM+1ΩM+1M+1)ω)
{n,n(1(2(1,,2(1,,2,,)2)2,,)2)2}
1219FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1ΩM+1M+1ΩM+1M+1+1)))) = ψ(ΩM+1ΩM+1M+1)ψΩM+1M+1)ω)
{n,n(1(1(1,,2(1,,2,,)2)1,2,,)2)2}
1220FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1M+1ΩM+1M+1ΩM+1M+1ΩM+1M+1+1))))) = ψ(ΩM+1ΩM+1M+1)ψΩM+1M+1)ψΩM+1M+1)ω)
{n,n(1(1(2(1,,2(1,,2,,)2)2(1,,2,,)2)2,,)2)2}
1221FGH ψ(ψM)
SAN
ψ(ΩM+1×2) = ψ(ΩM+1ΩM+1M+1ΩM+1M+1ΩM+1M+1+...)))) = ψ(ΩM+1ΩM+1M+1)ψΩM+1M+1)ψΩM+1M+1)ψΩM+1M+1)...)
{n,n(1(1,,2(1,,2,,)2)3(1,,2,,)2)2}
1222FGH ψ(ψM)
SAN
ψ(ΩM+12)
{n,n(1(1,,2(1,,2,,)2)1(1,,2(1,,2,,)2)2(1,,2,,)2)2}
1223FGH ψ(ψM)
SAN
ψ(ΩM+1ΩM+1)
{n,n(1(1(1,,2(1,,2,,)2)2,,2(1,,2,,)2)2(1,,2,,)2)2}
1224FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ΩM+2) = ψ(εΩM+1+1) = ψ(ΩM+1ΩM+1ΩM+1...)
ψ(2nd uncountable after 1st weakly Mahlo)
ψ(χM2(2))
{n,n(1,,3(1,,2,,)2)2}
1225FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ΩM+ω)
ψ(ω-th uncountable after 1st weakly Mahlo)
ψ(χM2(sup(n)))
{n,n(1,,1,2(1,,2,,)2)2}
1226FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩM+ω+1)
ψ(χM2(ω))
{n,n(1,,2,2(1,,2,,)2)2}
1227FGH ψ(ψM)
ψ(cardinal)
SAN
ψ(ΩM+Ω)
ψ(1st uncountable-th uncountable after 1st weakly Mahlo)
{n,n(1,,1'2(1,,2,,)2)2}
1228FGH ψ(ψM)
SAN
ψ(ΩM+ψχ(M)(M))
{n,n(1,,1(1,,1(1,,1,,2)2)2(1,,2,,)2)2}
1229FGH ψ(ψM)
SAN
ψ(ΩM+ψχ(M)M+1))
{n,n(1,,1(1,,1(1,,2(1,,2,,)2)2)2(1,,2,,)2)2}
1230FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩM+χ(M)) = ψ(ΩM+ψχ(M)M+ψχ(M)M+...)))
ψ(χM2(χ(M)))
{n,n(1,,1(1,,1,,2)2(1,,2,,)2)2}
1231FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩM+χ(MM))
ψ(χM2(χ(MM)))
{n,n(1,,1(1(1(1(1,,2,,)2)2,,)2)2(1,,2,,)2)2}
1232FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩM+ψχ(ΩM+1)M+1)) = ψ(ΩM+χ(MMMMM...))
ψ(χM2(χ(MMMMM...)))
{n,n(1,,1(1(1(1(1,,2(1,,2,,)2)2,,)2)2(1(1,,2(1,,2,,)2)2,,)2)2(1,,2,,)2)2}
1233FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩM+χ(ΩM+1)) = ψ(ΩM+ψχ(ΩM+1)M+ψχ(ΩM+1)M+...)))
ψ(χM2(χ(ΩM+1))) = ψ(χM2(χ(χM2(0))))
{n,n(1,,1(1(1(1,,2(1,,2,,)2)2,,)2)2(1,,2,,)2)2}
1234FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ΩM×2) = ψ(ΩM+M) = ψ(ΩM+χ(ΩM+χ(ΩM+...)))
ψ(1st weakly Mahlo-th uncountable after 1st weakly Mahlo)
ψ(χM2(M)) = ψ(χM2(χ(χM2(χ(χM2(χ(χM2(...))))))))
{n,n(1,,1(1(1,,2,,)2)2(1,,2,,)2)2}
1235FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ΩΩM×2)
ψ(1st weakly Mahlo-th uncountable after 1st weakly Mahlo-th uncountable after 1st weakly Mahlo)
ψ(χM2M2(M)))
{n,n(1,,1(1,,1(1(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1236FGH ψ(ψM)
ψ(cardinal)

FGH ψ(ψM2)
ψ(ψM2(cardinal))

FGH ψ(ψK)
SAN
ψ(ψIM+1(0)) = ψ(ΩΩΩ...ΩM+1)
ψ(1st fixed point of uncountable after 1st weakly Mahlo)
ψ(M2) = ψ(χM2M2M2M2(...χM2(M)...)))))
ψ(2nd weakly Mahlo)
ψ(χχK(2)(1,2)) = ψ(χK(1,2))
{n,n(1,,1,,2(1,,2,,)2)2}
1237FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1ΩM+1ΩΩ...ΩM+1))
ψ(M2ΩM+1(M2)) = ψ(M2χM2(1)(M2))
{n,n(1(1(1,,1,,2(1,,2,,)2)2,,)3)2}
1238FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1M+1)
ψ(M2M+1) = ψ(M2ΩM+1(M2ΩM+1(M2ΩM+1(M2+...)))) = ψ(M2M2(1))
{n,n(1(1,,2(1,,2,,)2)2(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1239FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1ΩM+2ΩΩ...ΩM+1))
ψ(M2ΩM+2(M2)) = ψ(M2χM2(2)(M2))
{n,n(1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)3(1,,2,,)2)2}
1240FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1ΩM+2ΩΩ...ΩM+1ΩM+2ΩΩ...ΩM+1)))
ψ(M2ΩM+2(M2ΩM+2(M2))) = ψ(M2χM2(2)(M2χM2(2)(M2)))
{n,n(1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1241FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1ΩM+2ΩΩ...ΩM+1ΩM+2ΩΩ...ΩM+1M+1)))
ψ(M2ΩM+2(M2ΩM+2(M2M+1))) = ψ(M2χM2(2)(M2χM2(2)(M2M2(1))))
{n,n(1(1,,2(1,,2,,)2)2,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1242FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1M+ω)
ψ(M2M+ω) = ψ(M2M2(sup(n)))
{n,n(1(1,,1,2(1,,2,,)2)2,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1243FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1M×2)
ψ(M2M×2) = ψ(M2M2(M))
{n,n(1(1,,1(1(1,,2,,)2)2(1,,2,,)2)2,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1244FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1×2) = ψ(ΩΩΩ...ΩM+1ΩΩ...ΩM+1)
ψ(M2ΩΩ...ΩM+1) = ψ(M2χM2(M2)(M2))
{n,n(1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1245FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1+1) = ψ(ΩψIM+1(0)+1)
ψ(M2M2χM2(M2)(M2)))
{n,n(1,,2(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1246FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1+2) = ψ(ΩψIM+1(0)+2)
ψ(M2M2χM2(M2)(M2)+1)
{n,n(1,,3(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1247FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1×2) = ψ(ΩψIM+1(0)×2)
ψ(M2M2χM2(M2)(M2)×2))
{n,n(1,,1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1248FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩΩΩ...ΩM+1+1) = ψ(ΩΩψIM+1(0)+1)
ψ(M2M2χM2(M2)(M2)+χM2χM2(M2)(M2))))
{n,n(1,,1,,2(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1249FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ψIM+1(2)) = ψ(ΩΩ...ψIM+1(0)+1)
ψ(2nd fixed point of uncountable after 1st weakly Mahlo)
ψ(M2χM2(M2)(M2+1)) = ψ(M2M2χM2(M2)(M2)+χM2χM2(M2)(M2)+χM2χM2(M2)(M2))+...)))
{n,n(1,,1(1,,1,,2(1,,2,,)2)3(1,,2,,)2)2}
1250FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ψIM+1IM+1(0)))
ψ(1st fixed point of uncountable after 1st weakly Mahlo-th fixed point of uncountable after 1st weakly Mahlo)
ψ(M2χM2(M2)(M2χM2(M2)(M2)))
{n,n(1,,1(1,,1,,2(1,,2,,)2)1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1251FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(IM+1) = ψ(I(1,M+1)) = ψ(ψIM+1IM+1IM+1(...))))
ψ(1st weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2(M2)) = ψ(M2χM2(M2)(M2χM2(M2)(M2χM2(M2)(M2+...)))))
{n,n(1,,1(1,,1,,2(1,,2,,)2)1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1252FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(IM+1IM+1IM+1IM+1...) = ψ(εIM+1+1 ) = ψ(ΩIM+1) = ψ(I(0,I(1,M+1)+1))
ψ(1st uncountable after 1st weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2(M2+1)) = ψ(M2M2(M2)χM2(M2)χM2(M2)χM2(M2)...)
{n,n(1,,2,,2(1,,2,,)2)2}
1253FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ΩIM+1×2)
ψ(M2M2(M2M2(M2)))
{n,n(1,,1(1,,1,,2(1,,2,,)2)2,,2(1,,2,,)2)2}
1254FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ψIM+2(0)) = ψ(ΩΩ...ΩIM+1)
ψ(1st fixed point of uncountable after 1st weakly inaccessible after 1st weakly Mahlo)
ψ(M2×2) = ψ(M2+M2) = ψ(M2M2(M2M2(M2M2(M2+...))))
{n,n(1,,1,,3(1,,2,,)2)2}
1255FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ψIM+2IM+2(0)) = ψ(IM+Ω...ΩIM+1(0))
ψ(1st fixed point of uncountable after 1st weakly inaccessible after 1st weakly Mahlo-th fixed point of uncountable after 1st weakly inaccessible after 1st weakly Mahlo)
ψ(M2×2+ψχM2(M2×2)(M2×2))
{n,n(1,,1(1,,1,,3(1,,2,,)2)1(1,,1,,3(1,,2,,)2)2(1,,2,,)2)2}
1256FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(IM+2) = ψ(I(1,M+2)) = ψ(ψIM+2IM+2IM+2(...))))
ψ(2nd weakly inaccessible after 1st weakly Mahlo)
ψ(M2×2+χM2(M2×2)) = ψ(M2×2+ψχM2(M2×2)(M2×2+ψχM2(M2×2)(M2×2+...))))
{n,n(1,,1(1,,1,,3(1,,2,,)2)1(1,,1,,3(1,,2,,)2)2(1,,2,,)2)2}
1257FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(ψIM×2(0))
ψ(1st weakly Mahlo-th weakly inaccessible after 1st weakly Mahlo)
ψ(M2×M)
{n,n(1,,1,,1(1(1,,2,,)2)2(1,,2,,)2)2}
1258FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(IIM+1) = ψ(I(1,I(1,M+1)+1))
ψ(1st weakly inaccessible after 1st weakly Mahlo-th weakly inaccessible after 1st weakly Mahlo)
ψ(M2×χM2(M2))
{n,n(1,,1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1259FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(2,M+1)(0)) = ψ(III...IM+1) = ψ(I(1,I(1,I(1,I(1,...I(1,M+1)+1...)))))
ψ(1st fixed point of weakly inaccessible after 1st weakly Mahlo)
ψ(M22)
{n,n(1,,1,,1,,2(1,,2,,)2)2}
1260FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(2,M+1)) = ψ(ψI(2,M+1)I(2,M+1)I(2,M+1)I(2,M+1)(...)))))
ψ(1st 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22M2(M22))
{n,n(1,,1(1,,1,,1,,2(1,,2,,)2)1(1,,1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1261FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ΩI(2,M+1)) = ψ(I(0,I(2,M+1)+1)) = ψ(εI(2,M+1)+1) = ψ(I(2,M+1)I(2,M+1)I(2,M+1)I(2,M+1)...)
ψ(1st uncountable after 1st 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22M2(M22+1))
{n,n(1,,1,,2,,2(1,,2,,)2)2}
1262FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(1,I(2,M+1)+1)(0)) = ψ(ψII(2,M+1)+1(0)) = ψ(ΩΩΩ...I(2,M+1)+1) = ψ(I(0,I(0,I(0,I(0,...I(2,M+1)+1))))
ψ(1st fixed point of uncountable after 1st 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22+M2)
{n,n(1,,1,,2,,2(1,,2,,)2)2}
1263FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(II(2,M+1)+1) = ψ(I(1,I(2,M+1)+1)) = ψ(ψI(1,I(2,M+1)+1)I(1,I(2,M+1)+1)(...)))
ψ(1st weakly inaccessible after 1st 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22+M2M2(M22+M2))
{n,n(1,,1(1,,1,,2,,2(1,,2,,)2)1(1,,1,,2,,2(1,,2,,)2)2(1,,2,,)2)2}
1264FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(2,M+2)(0)) = ψ(I(1,I(1,I(1,I(1,...I(2,M+1)+1...))))) = ψ(III...I(2,M+1)+1)
ψ(1st fixed point of weakly inaccessible after 1st 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22×2)
{n,n(1,,1,,1,,3(1,,2,,)2)2}
1265FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(2,M+2)) = ψ(ψI(2,M+2)I(2,M+2)I(2,M+2)I(2,M+2)(...)))))
ψ(2nd 2-weakly inaccessible after 1st weakly Mahlo)
ψ(M22×2+χM2(M22×2))
{n,n(1,,1(1,,1,,1,,3(1,,2,,)2)1(1,,1,,1,,3(1,,2,,)2)2(1,,2,,)2)2}
1266FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(3,M+1)(0)) = ψ(I(2,I(2,I(2,I(2,...I(2,M+1)+1...)))))
ψ(1st fixed point of 2-weakly inaccessible after 1st weakly Mahlo
ψ(M23)
{n,n(1,,1,,1,,1,,2(1,,2,,)2)2}
1267FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(3,M+1)) = ψ(ψI(3,M+1)I(3,M+1)I(3,M+1)I(3,M+1)(...)))))
ψ(1st 3-weakly inaccessible after 1st weakly Mahlo)
ψ(M22M2(M22))
{n,n(1,,1(1,,1,,1,,1,,2(1,,2,,)2)1(1,,1,,1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1268FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(ω,M+1)(0)) = ψ(I(sup(n),M+1))
ψ(1st fixed point of n-weakly inaccessible after 1st weakly Mahlo)
ψ(M2ω)
{n,n(1(2,,)2(1,,2,,)2)2}
1269FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(ω,M+1)) = ψ(ψI(ω,M+1)I(ω,M+1)I(ω,M+1)I(ω,M+1)(...)))))
ψ(1st ω-weakly inaccessible after 1st weakly Mahlo)
ψ(M2ωM2(M2ω))
{n,n(1,,1(1(2,,)2(1,,2,,)2)1(1(2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1270FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ψI(I,M+1)(0)) = ψ(ψI(χ(M),M+1)(0))
ψ(M2χ(M2))
{n,n(1(1(1,,1,,2)2,,)2(1,,2,,)2)2}
1271FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(I(χ(M),M+1)) = ψ(ψI(χ(M),M+1)I(χ(M),M+1)I(χ(M),M+1)(...))))
ψ(1st weakly inaccessible-weakly inaccessible after 1st weakly Mahlo)
ψ(M2χ(M)M2(M2χ(M)))
{n,n(1,,1(1(1(1,,1,,2)2,,)2(1,,2,,)2)1(1(1(1,,1,,2)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1272FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ψI(χ(M2),M+1)(0)) = ψ(ψI(I(2,0),M+1)(0))
ψ(M2χ(M2))
{n,n(1(1(1,,1,,1,,2)2,,)2(1,,2,,)2)2}
1273FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)
SAN
ψ(I(I(2,0),M+1)) = ψ(ψI(χ(M2),M+1)I(χ(M2),M+1)I(χ(M2),M+1)(...))))
ψ(1st 2-weakly inaccessible-weakly inaccessible after 1st weakly Mahlo)
ψ(M2χ(M2)M2(M2χ(M2)))
{n,n(1,,1(1(1(1,,1,,1,,2)2,,)2(1,,2,,)2)1(1(1(1,,1,,1,,2)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1274FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ψI(χ(I(1,M+1)),M+1)(0))
ψ(M2χ(χM2(M2)))
{n,n(1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2}
1275FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(χ(I(1,M+1)),M+1)) = ψ(ψI(χ(I(1,M+1)),M+1)I(χ(I(1,M+1)),M+1)I(χ(I(1,M+1)),M+1)(...))))
ψ(M2χ(χM2(M2))M2(M2χ(χM2(M2))))
{n,n(1,,1(1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)1(1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1,,2,,)2)2}
1276FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ψI(M,1)(0)) = ψ(ψI(M,M+1)(0)) = ψ(I(χ(I(χ(I(χ(I(...,M+1)),M+1)),M+1)),M+1))
ψ(M2M) = ψ(M2χ(χM2(M2χ(χM2(M2...)))))
{n,n(1(1(1(1,,2,,)2)2,,)2(1,,2,,)2)2}
1277FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(M,1)) = ψ(I(M,M+1)) = ψ(ψI(M,1)I(M,1)I(M,1)(...))))
ψ(1st weakly Mahlo-weakly inaccessible after 1st weakly Mahlo)
ψ(M2MM2(M2M))
{n,n(1,,1(1(1(1,,2,,)2)2,,)2(1,,2,,)2)1(1(1(1,,2,,)2)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1278FGH ψ(ψM)
FGH ψ(ψM2)
SAN
ψ(ψI(I(M,1),0)(0))
ψ(M2χM2(M2))
{n,n(1(1(1,,1,,2(1,,2,,)2)2,,)2(1,,2,,)2)2}
1279FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(I(M,1),0))
ψ(1st weakly inaccessible after weakly Mahlo-weakly inaccessible after 1st weakly Mahlo)
ψ(M2χM2(M2)M2(M2χM2(M2)))
{n,n(1,,1(1(1(1,,1,,2(1,,2,,)2)2,,)2(1,,2,,)2)1(1(1(1,,1,,2(1,,2,,)2)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1280FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)) = ψ(I(I(I(...I(M,1),...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible after 1st weakly Mahlo))
ψ(M2M2)
{n,n(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2} = {n,n(1(1,,2,,)3)2}
1281FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+ψΩM+1I(1,0,M+1)(0)))
ψ(M2M2ΩM+1(M2M2)) = ψ(M2M2χM2(1)(M2M2))
{n,n(1(1(1(1,,2,,)3)2,,)3)2}
1282FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+ΩM+1) = ψ(ψI(1,0,M+1)(0)+ψΩM+1I(1,0,M+1)(0)+ψΩM+1I(1,0,M+1)(0)+...)))
ψ(M2M2M+1) = ψ(M2M2ΩM+1(M2M2ΩM+1(M2M2ΩM+1(M2M2+...))))
{n,n(1(1,,2(1,,2,,)2)2(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1283FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+ΩM+ω)
ψ(M2M2M+ω)
{n,n(1(1,,1,2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1284FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+ψI(1,M+1)(0)) = ψ(ψI(1,0,M+1)(0)+ΩΩΩ...ΩM+1)
ψ(M2M2χM2(M2)(M2)) = ψ(M2M2ΩΩ...ΩM+1)
{n,n(1(1,,1(1,,1,,2(1,,2,,)2)2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1285FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+ψI(1,M+1)I(1,0,M+1)(0)))
ψ(M2M2χM2(M2)(M2M2))
{n,n(1(1,,1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1286FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+I(1,M+1)) = ψ(ψI(1,0,M+1)(0)+ψI(1,M+1)I(1,0,M+1)(0)+ψI(1,M+1)I(1,0,M+1)(0)+...)))
ψ(M2M2M2(M2)) = ψ(M2M2χM2(M2)(M2M2χM2(M2)(M2M2+...)))
{n,n(1(1,,1,,2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1287FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+I(1,M+2))
ψ(M2M2M2(M2×2))
{n,n(1(1,,1,,3(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1288FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)+I(2,M+1))
ψ(M2M2M2(M22))
{n,n(1(1,,1,,1,,2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1289FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(0)×2) = ψ(ψI(1,0,M+1)(0)+ψI(1,0,M+1)(0))
ψ(M2M2χM2(M2M2)(M2M2)) = ψ(M2M2I(1,0,M+1)(0))
{n,n(1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1290FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(I(ψI(1,0,M+1)(0),M+1))
ψ(M2M2M2(M2ψχM2(M2M2)(M2M2)))
{n,n(1,,2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1291FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ΩI(ψI(1,0,M+1)(0),M+1))+1)
ψ(M2M2M2(M2ψχM2(M2M2)(M2M2)+1))
{n,n(1,,3(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1292FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(IM+I(ψI(1,0,M+1)(0),M+1))+1)
ψ(M2M2M2(M2ψχM2(M2M2)(M2M2)+M2))
{n,n(1,,1,,2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1293FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(I(ψI(1,0,M+1)(0),M+2))
ψ(M2M2M2(M2ψχM2(M2M2)(M2M2)×2))
{n,n(1(1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2,,)2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1294FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(I(ψI(1,0,M+1)(0)+1,0))
ψ(M2M2M2(M2ψχM2(M2M2)(M2M2)+1))
{n,n(1(1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2,,)1,,2(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1295FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+1)(1)) = ψ(I(I(I(I(...I(ψI(1,0,M+1)(0),1)...,0),0),0),0))
ψ(2nd fixed point of α-weakly inaccessible after 1st weakly Mahlo))
ψ(M2M2χM2(M2M2)(M2M2+1)) = ψ(M2M2M2(M2χM2(M2χM2(M2...ψχM2(M2M2)(M2M2)))))
{n,n(1(1(1(1,,2,,)3)2,,)3(1,,2,,)2)2}
1296FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(1,0,M+1)) = ψ(ψI(1,0,M+1)I(1,0,M+1)I(1,0,M+1)(...))))
ψ(hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2M2(M2M2)) = ψ(M2M2χM2(M2M2)(M2M2χM2(M2M2)(M2M2χM2(M2M2)(M2M2+...))))
{n,n(1(1(1(1,,2,,)3)2,,)1,,2(1,,2,,)2)2}
1297FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ΩI(1,0,M+1))+1)
ψ(1st uncountable after hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2M2(M2M2+1))
{n,n(1,,2(1(1(1,,2,,)3)2,,)1,,2(1,,2,,)2)2}
1298FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(ψI(1,I(1,0,M+1)+1)(0)) = ψ(ΩΩΩ...ΩI(1.0M+1)+1)
ψ(1st fixed point of uncountable after hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2+M2) = ψ(M2M2M2(M2M2M2(M2M2M2(M2M2+...))))
{n,n(1,,1,,2(1(1(1,,2,,)3)2,,)1,,2(1,,2,,)2)2}
1299FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+2)(0))
ψ(M2M2×2)
{n,n(1(1(1(1,,2,,)3)2,,)2,,2(1,,2,,)2)2}
1300FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(1,0,M+2)) = ψ(ψI(1,0,M+2)I(1,0,M+2)I(1,0,M+2)(...))))
ψ(2nd hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2×2+χM2(M2M2×2))
{n,n(1(1(1(1,,2,,)3)2,,)1,,3(1,,2,,)2)2}
1301FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,M+ω)(0))
ψ(M2M2×ω)
{n,n(1(1(1(1,,2,,)3)2,,)1,,1,2(1,,2,,)2)2}
1302FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,1,M+1)(0)) = ψ(I(1,0,I(1,0,I(1,0,...I(1,0,M+1)...))))
ψ(M2M2+1)
{n,n(1(1(1(1,,2,,)3)2,,)1,,1,,2(1,,2,,)2)2}
1303FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(1,1,M+1)) = ψ(ψI(1,1,M+1)I(1,0,M+1)I(1,0,M+1)(...))))
ψ(2-hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2+1M2(M2M2+1))
{n,n(1(1(1(1,,2,,)3)2,,)1,,1,,3(1,,2,,)2)2}
1304FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(2,0,M+1)(0)) = ψ(I(1,I(1,I(1,I(1,...I(1,1,M+1)..,0),0),0),0))
ψ(M2M2×2)
{n,n(1(1(1(1,,2,,)3)2,,)1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1305FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(2,0,M+1)) = ψ(ψI(2,0,M+1)I(2,0,M+1)I(2,0,M+1)(...))))
ψ(hyper-hyper-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2×2M2(M2M2×2))
{n,n(1,,1(1(1(1(1,,2,,)3)2,,)1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)1(1(1(1(1,,2,,)3)2,,)1(1(1(1,,2,,)3)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1306FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(ω,0,M+1)(0)) = ψ(I(sup(n),0,M+1))
ψ(M2M2×ω)
{n,n(1(2(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1307FGH ψ(ψM)
ψ(cardinal)
FGH ψ(ψM2)

SAN
ψ(I(ω,0,M+1)) = ψ(ψI(ω,0,M+1)I(ω,0,M+1)I(ω,0,M+1)(...))))
ψ(hyperω-weakly inaccessible after 1st weakly Mahlo)
ψ(M2M2×ωM2(M2M2×ω))
{n,n(1,,1(1(2(1(1,,2,,)3)2,,)2(1,,2,,)2)1(1(2(1(1,,2,,)3)2,,)2(1,,2,,)2)2(1,,2,,)2)2}
1308FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,0,M+1)(0))
ψ(M2M22)
{n,n(1(1(1(1,,2,,)3)3,,)2(1,,2,,)2)2}
1309FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1,0,0,0,0,...M+1)(0)) = ψ(ψI(1ωM+10)(0))
ψ(M2M2ω)
{n,n(1(1(1(1,,2,,)3)1,2,,)2(1,,2,,)2)2}
1310FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1MM+10)(0))
ψ(M2M2M)
{n,n(1(1(1(1,,2,,)3)1(1(1,,2,,)2)2,,)2(1,,2,,)2)2}
1311FGH ψ(ψM)
FGH ψ(ψM2)

SAN
ψ(ψI(1I(1I(1...I(1MM+10)...)))(0))
ψ(M2M2M2)
{n,n(1(1(1(1,,2,,)3)1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1312FGH ψ(ψM2)
SAN
ψ(M2M2M2M2)
{n,n(1(1(1,,2,,)3)2(1,,2,,)3)2}
1313FGH ψ(ψM2)
SAN
ψ(M2M2M2M2M2)
{n,n(1(1(1,,2,,)3)1(1(1,,2,,)3)2(1,,2,,)3)2}
1314FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(M2M2M2M2M2...) = ψ(εM2+1) = ψ(ΩM2+1)
ψ(1st uncountable after 2nd weakly Mahlo)
ψ(χM3(1))
{n,n(1,,2(1,,2,,)3)2}
1315FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ΩM2+2) = ψ(εΩM2+1+1) = ψ(ΩM2+1ΩM2+1ΩM2+1...)
ψ(2nd uncountable after 2nd weakly Mahlo)
ψ(χM3(2))
{n,n(1,,3(1,,2,,)3)2}
1316FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ΩM2×2)
ψ(2nd weakly Mahlo-th uncountable after 2nd weakly Mahlo)
ψ(χM3(M2))
{n,n(1,,1(1(1,,2,,)3)2(1,,2,,)3)2}
1317FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
ψ(ψM3(cardinal))
FGH ψ(ψK)
SAN
ψ(ΩΩΩ...ΩM2+1) = ψ(ψIM2+1(0)) = ψ(ψIψIM+1(0)+1(0))
ψ(1st fixed point of uncountable after 2nd weakly Mahlo)
ψ(M3) = ψ(χM3M3M3M3(...χM3(M2)...)))))
ψ(3d weakly Mahlo)
ψ(χχK(2)(1,3))
{n,n(1,,1,,2(1,,2,,)3)2}
1318FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψIM2+1(1)) = ψ(ΩΩΩ...ΩM2+1+1) = ψ(ΩΩ...ψIM2+1(0)+1)
ψ(2nd fixed point of uncountable after 2nd weakly Mahlo)
ψ(M3χM3(M3)(M3+1))
{n,n(1,,1(1,,1,,2(1,,2,,)3)3(1,,2,,)3)2}
1319FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(IM2+1) = ψ(I(1,M2+1)) = ψ(ψIM2+1IM2+1IM2+1(...))))
ψ(1st weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3(M3)) = ψ(M3χM3(M3)(M3χM3(M3)(M3χM3(M3)(M3+...)))))
{n,n(1,,1(1,,1,,2(1,,2,,)3)1(1,,1,,2(1,,2,,)3)2(1,,2,,)3)2}
1320FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(IM2+1IM2+1IM2+1IM2+1...) = ψ(εIM2+1+1 ) = ψ(ΩIM2+1) = ψ(I(0,I(1,M2+1)+1))
ψ(1st uncountable after 1st weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3(M3+1)) = ψ(M3M3(M3)χM3(M3)χM3(M3)χM3(M3)...)
{n,n(1,,2,,2(1,,2,,)3)2}
1321FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψIM2+2(0)) = ψ(ΩΩ...ΩIM2+1)
ψ(1st fixed point of uncountable after 1st weakly inaccessible after 2nd weakly Mahlo)
ψ(M3×2)
{n,n(1,,1,,3(1,,2,,)3)2}
1322FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(IM2+2) = ψ(I(1,M2+2)) = ψ(ψIM2+2IM2+2IM2+2(...))))
ψ(2nd weakly inaccessible after 2nd weakly Mahlo)
ψ(M3×2+χM3(M3×2))
{n,n(1,,1(1,,1,,3(1,,2,,)3)1(1,,1,,3(1,,2,,)3)2(1,,2,,)3)2}
1323FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψI(2,M2+1)(0)) = ψ(III...IM2+1) = ψ(I(1,I(1,I(1,I(1,...I(1,M2+1)+1...)))))
ψ(1st fixed point of weakly inaccessible after 2nd weakly Mahlo)
ψ(M32)
{n,n(1,,1,,1,,2(1,,2,,)3)2}
1324FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(2,M2+1)) = ψ(ψI(2,M2+1)I(2,M2+1)I(2,M2+1)I(2,M2+1)(...)))))
ψ(1st 2-weakly inaccessible after 2nd weakly Mahlo)
ψ(M32M3(M32))
{n,n(1,,1(1,,1,,1,,2(1,,2,,)3)1(1,,1,,1,,2(1,,2,,)3)2(1,,2,,)3)2}
1325FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψI(ω,M2+1)(0)) = ψ(I(sup(n),M2+1))
ψ(1st fixed point of n-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3ω)
{n,n(1(2,,)2(1,,2,,)3)2}
1326FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(ω,M2+1)) = ψ(ψI(ω,M2+1)I(ω,M2+1)I(ω,M2+1)I(ω,M2+1)(...)))))
ψ(1st ω-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3ωM3(M3ω))
{n,n(1,,1(1(2,,)2(1,,2,,)3)1(1(2,,)2(1,,2,,)3)2(1,,2,,)3)2}
1327FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(M,M2+1)(0)) = ψ(I(χ(I(χ(I(χ(I(...,M2+1)),M2+1)),M2+1)),M2+1))
ψ(M3M) = ψ(M3χ(χM3(M3χ(χM3(M3...)))))
{n,n(1(1(1(1,,2,,)2)2,,)2(1,,2,,)3)2}
1328FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(M,M2+1)) = ψ(ψI(M,M2+1)I(M,M2+1)I(M,M2+1)I(M,M2+1)(...)))))
ψ(1st weakly Mahlo-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3MM3(M3M))
{n,n(1,,1(1(1(1(1,,2,,)2)2,,)2(1,,2,,)3)1(1(1(1(1,,2,,)2)2,,)2(1,,2,,)3)2(1,,2,,)3)2}
1329FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(M2,1)(0)) = ψ(ψI(M2,M2+1)(0)) = ψ(I(χM2(I(χM2(I(χM2(I(...,M2+1)),M2+1)),M2+1)),M2+1))
ψ(M3M2) = ψ(M3χM2M3(M3χM2M3(M3...)))))
{n,n(1(1(1(1,,2,,)3)2,,)2(1,,2,,)3)2}
1330FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(M2,1)) = ψ(I(M2,M2+1)) = ψ(ψI(M2,1)I(M2,1)I(M2,1)(...))))
ψ(2nd weakly Mahlo-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M2M3(M3M2))
{n,n(1,,1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)3)1(1(1(1(1,,2,,)3)2,,)2(1,,2,,)3)2(1,,2,,)3)2}
1331FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψI(1,0,M2+1)(0)) = ψ(I(I(I(...I(M2,1),...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible after 2nd weakly Mahlo))
ψ(M3M3)
{n,n(1(1(1(1,,2,,)4)2,,)2(1,,2,,)3)2} = {n,n(1(1,,2,,)4)2}
1332FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψI(1,0,M2+1)(1)) = ψ(I(I(I(I(...I(ψI(1,0,M2+1)(0),1)...,0),0),0),0))
ψ(2nd fixed point of α-weakly inaccessible after 2nd weakly Mahlo))
ψ(M3M3χM3(M3M3)(M3M3+1)) = ψ(M3M3M3(M3χM3(M3χM3(M3...ψχM3(M3M3)(M3M3)))))
{n,n(1(1(1(1,,2,,)4)2,,)3(1,,2,,)3)2}
1333FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(1,0,M2+1)) = ψ(ψI(1,0,M2+1)I(1,0,M2+1)I(1,0,M2+1)(...))))
ψ(hyper-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3M3(M3M3)) = ψ(M3M3χM3(M3M3)(M3M3χM3(M3M3)(M3M3χM3(M3M3)(M3M3+...))))
{n,n(1(1(1(1,,2,,)4)2,,)1,,2(1,,2,,)3)2}
1334FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ΩI(1,0,M2+1))+1)
ψ(1st uncountable after hyper-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3M3(M3M3+1))
{n,n(1,,2(1(1(1,,2,,)4)2,,)1,,2(1,,2,,)3)2}
1335FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(ψI(1,I(1,0,M2+1)+1)(0)) = ψ(ΩΩΩ...ΩI(1.0M2+1)+1)
ψ(1st fixed point of uncountable after hyper-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3+M3) = ψ(M3M3M3(M3M3M3(M3M3M3(M3M3+...))))
{n,n(1,,1,,2(1(1(1,,2,,)4)2,,)1,,2(1,,2,,)3)2}
1336FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1,0,M2+2)(0))
ψ(M3M3×2)
{n,n(1(1(1(1,,2,,)4)2,,)2,,2(1,,2,,)3)2}
1337FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(1,0,M2+2)) = ψ(ψI(1,0,M2+2)I(1,0,M2+2)I(1,0,M2+2)(...))))
ψ(2nd hyper-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3×2+χM3(M3M3×2))
{n,n(1(1(1(1,,2,,)3)2,,)1,,3(1,,2,,)2)2}
1338FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1,1,M2+1)(0))
ψ(M3M3+1)
{n,n(1(1(1(1,,2,,)4)2,,)1,,1,,2(1,,2,,)3)2}
1339FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(1,1,M2+1)) = ψ(ψI(1,1,M2+1)I(1,0,M2+1)I(1,0,M2+1)(...))))
ψ(2-hyper-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3+1M3(M3M3+1))
{n,n(1(1(1(1,,2,,)4)2,,)1,,1,,3(1,,2,,)2)2}
1340FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(ω,0,M+1)(0)) = ψ(I(sup(n),0,M+1))
ψ(M3M3×ω)
{n,n(1(2(1(1,,2,,)4)2,,)2(1,,2,,)3)2}
1341FGH ψ(ψM2)
ψ(cardinal)
FGH ψ(ψM3)
SAN
ψ(I(ω,0,M+1)) = ψ(ψI(ω,0,M+1)I(ω,0,M+1)I(ω,0,M+1)(...))))
ψ(hyperω-weakly inaccessible after 2nd weakly Mahlo)
ψ(M3M3×ωM3(M3M3×ω))
{n,n(1,,1(1(2(1(1,,2,,)4)2,,)2(1,,2,,)3)1(1(2(1(1,,2,,)4)2,,)2(1,,2,,)3)2(1,,2,,)2)2}
1342FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1,0,0,M+1)(0))
ψ(M3M32)
{n,n(1(1(1(1,,2,,)4)3,,)2(1,,2,,)3)2}
1343FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1,0,0,0,0,...M+1)(0)) = ψ(ψI(1ωM+10)(0))
ψ(M3M3ω)
{n,n(1(1(1(1,,2,,)4)1,2,,)2(1,,2,,)2)2}
1344FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1MM2+10)(0))
ψ(M3M3M)
{n,n(1(1(1(1,,2,,)4)1(1(1,,2,,)2)2,,)2(1,,2,,)2)2}
1345FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1M2M2+10)(0))
ψ(M3M3M2)
{n,n(1(1(1(1,,2,,)4)1(1(1,,2,,)3)2,,)2(1,,2,,)2)2}
1346FGH ψ(ψM2)
FGH ψ(ψM3)
SAN
ψ(ψI(1I(1I(1...I(1M2M2+10)...)))(0))
ψ(M3M3M3)
{n,n(1(1(1(1,,2,,)4)1(1(1,,2,,)4)2,,)2(1,,2,,)3)2}
1347FGH ψ(ψM3)
SAN
ψ(M3M3M3M3)
{n,n(1(1(1,,2,,)4)2(1,,2,,)4)2}
1348FGH ψ(ψM3)
SAN
ψ(M3M3M3M3M3)
{n,n(1(1(1,,2,,)4)1(1(1,,2,,)4)2(1,,2,,)4)2}
1349FGH ψ(ψM3)
ψ(cardinal)
FGH ψ(ψM4)
SAN
ψ(M3M3M3M3M3...) = ψ(εM3+1) = ψ(ΩM3+1)
ψ(1st uncountable after 3d weakly Mahlo)
ψ(χM4(1))
{n,n(1,,2(1,,2,,)4)2}
1350FGH ψ(ψM3)
ψ(cardinal)
FGH ψ(ψM4)
ψ(ψM4(cardinal))

SAN
ψ(ΩΩΩ...ΩM3+1) = ψ(ψIM3+1(0)) = ψ(ψIψIψIM+1(0)+1(0)+1(0))
ψ(1st fixed point of uncountable after 3d weakly Mahlo)
ψ(M4) = ψ(χM4M4M4M4(...χM4(M3)...)))))
ψ(4th weakly Mahlo)
{n,n(1,,1,,2(1,,2,,)4)2}
1351FGH ψ(ψM3)
ψ(cardinal)
FGH ψ(ψM4)
SAN
ψ(ψI(1,0,M3+1)(0)) = ψ(I(I(I(...I(M3,1),...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible after 3d weakly Mahlo))
ψ(M4M4)
{n,n(1(1,,2,,)5)2}
1352FGH ψ(ψM4)
ψ(cardinal)
FGH ψ(ψM5)
ψ(ψM5(cardinal))

SAN
ψ(ΩΩΩ...ΩM4+1) = ψ(ψIM4+1(0)) = ψ(ψIψIψIψIM+1(0)+1(0)+1(0)+1(0))
ψ(1st fixed point of uncountable after 4-th weakly Mahlo)
ψ(M5) = ψ(χM5M5M5M5(...χM5(M4)...)))))
ψ(5-th weakly Mahlo)
{n,n(1,,1,,2(1,,2,,)5)2}
1353FGH ψ(ψM4)
ψ(cardinal)
FGH ψ(ψM5)
SAN
ψ(ψI(1,0,M4+1)(0)) = ψ(I(I(I(...I(M4,1),...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible after 4-th weakly Mahlo))
ψ(M5M5)
{n,n(1(1,,2,,)6)2}
1354FGH ψ-function
ψ(cardinal)

SAN
ψ(Mω) = ψ(Msup(n)) = ψ(ψIψIψI...ψIM+1(0)+1...(0)+1(0)+1(0))
ψ(ω-th weakly Mahlo)
{n,n(1(1,,2,,)1,2)2}
1355FGH ψ-function
SAN
ψ(MωΩM+1(Mω))
{n,n(1(1(1(1,,2,,)1,2)2,,)3)2}
1356FGH ψ-function
SAN
ψ(MωΩM+1(Mω)×ω)
{n,n(1(1(1(1,,2,,)1,2)2,,)1,2)2}
1357FGH ψ-function
SAN
ψ(MωΩM+1(Mω)ω)
{n,n(1(2(1(1,,2,,)1,2)2,,)2)2}
1358FGH ψ-function
SAN
ψ(MωΩM+1(Mω)ψΩM+1(Mω)ω)
{n,n(1(1(1(1,,2,,)1,2)1,2,,)2)2}
1359FGH ψ-function
SAN
ψ(MωΩM+1(Mω)ψΩM+1(Mω)ψΩM+1(Mω)ω)
{n,n(1(1(2(1,,2,,)1,2)2,,)2)2}
1360FGH ψ-function
SAN
ψ(MωΩM+1(Mω)ψΩM+1(Mω)ψΩM+1(Mω)M)
{n,n(1(1(1,,2,,)2)2(1,,2,,)1,2)2}
1361FGH ψ-function
SAN
ψ(MωΩM2+1(Mω))
{n,n(1(1(1(1,,2,,)1,2)2,,)3(1,,2,,)2)2}
1362FGH ψ-function
SAN
ψ(MωΩM+2(Mω)ψΩM+2(Mω)ψΩM+2(Mω)M2)
{n,n(1(1(1,,2,,)3)2(1,,2,,)1,2)2}
1363FGH ψ-function
SAN
ψ(MωΩM3+1(Mω))
{n,n(1(1(1(1,,2,,)1,2)2,,)4(1,,2,,)2)2}
1364FGH ψ-function
SAN
ψ(MωΩM+3(Mω)ψΩM+3(Mω)ψΩM+3(Mω)M3)
{n,n(1(1(1,,2,,)4)2(1,,2,,)1,2)2}
1365FGH ψ-function
SAN
ψ(Mω×2)
{n,n(1(1(1,,2,,)1,2)2(1,,2,,)1,2)2}
1366FGH ψ-function
SAN
ψ(MωMωMωMωMω...) = ψ(εMω+1) = ψ(ΩMω+1) = ψ(χMω+1(1))
{n,n(1,,2(1,,2,,)1,2)2}
1367FGH ψ-function
SAN
ψ(ΩMω+2) = ψ(εΩMω+1+1) = ψ(ΩMω+1ΩMω+1ΩMω+1...) = ψ(χMω+1(2))
{n,n(1,,3(1,,2,,)1,2)2}
1368FGH ψ-function
SAN
ψ(Mω+1) = ψ(χMω+1Mω+1Mω+1Mω+1(...χMω+1(Mω)...)))))
{n,n(1,,1,,2(1,,2,,)1,2)2}
1369FGH ψ-function
SAN
ψ(Mω+1Mω+1)
{n,n(1(1,,2,,)2,2)2}
1370FGH ψ-function
SAN
ψ(Mω+1Mω+1Mω+1Mω+1Mω+1...) = ψ(εMω+1+1) = ψ(ΩMω+1+1) = ψ(χMω+2(1))
{n,n(1,,2(1,,2,,)2,2)2}
1371FGH ψ-function
SAN
ψ(Mω+2) = ψ(χMω+2Mω+2Mω+2Mω+2(...χMω+2(Mω+1)...)))))
{n,n(1,,1,,2(1,,2,,)2,2)2}
1372FGH ψ-function
SAN
ψ(Mω×2)
{n,n(1(1,,2,,)1,3)2}
1373FGH ψ-function
SAN
ψ(Mε0) = ψ(Mψ(Ω)))
{n,n(1(1,,2,,)1(1'2)2)2}
1374FGH ψ-function
ψ(cardinal)

SAN
ψ(MΩ) = ψ(Mψ(Mψ(Mψ(Mψ(M...)))))
ψ(1st uncountable ordinal-th weakly Mahlo)
{n,n(1(1,,2,,)1'2)2}
1375FGH ψ-function
ψ(cardinal)

SAN
ψ(MΩ2) = ψ(Mψ1(Mψ1(Mψ1(...))))
ψ(2nd uncountable ordinal-th weakly Mahlo)
{n,n(1(1,,2,,)1(1,,3)2)2}
1376FGH ψ-function
ψ(cardinal)

SAN
ψ(MΩΩΩ...) = ψ(Mψχ(M)(M))
ψ(1st fixed point of uncountable ordinal-th weakly Mahlo)
{n,n(1(1,,2,,)1(1,,1(1,,1,,2)2)2)2}
1377FGH ψ-function
ψ(cardinal)

SAN
ψ(Mχ(M)) = ψ(MI) =ψ(Mψχ(M)(Mψχ(M)(Mψχ(M)(Mψχ(M)(M...)))))
ψ(1st weakly inaccessible-th weakly Mahlo)
{n,n(1(1,,2,,)1(1,,1,,2)2)2}
1378FGH ψ-function
ψ(cardinal)

SAN
ψ(MM) = ψ(Mχ(Mχ(Mχ(M...))))
ψ(1st weakly Mahlo-th weakly Mahlo)
{n,n(1(1,,2,,)1(1(1,,2,,)2)2)2}
1379FGH ψ-function
ψ(cardinal)

SAN
ψ(MMM)
ψ(1st weakly Mahlo-th weakly Mahlo-th weakly Mahlo)
{n,n(1(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)2)2)2)2}
1380FGH ψ-function
ψ(cardinal)

SAN
ψ(MMMM)
ψ(Mahlo-th weakly Mahlo-th weakly Mahlo-th weakly Mahlo)
{n,n(1(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)2)2)2)2)2}
1381FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)) = ψ(MMMM...) = ψ(ψM(0,α)-I(1,0)(0)) = ψ(M(0,M(0,M(0,M(0,...)))))
ψ(1st fixed point of weakly Mahlo)
ψ(ψχM(1,0)(0)(0))
ψ(ψχχ1M(2;0)(0)(0)(0)) = ψ(M(1;M(1;M(1;M(1;...)))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)2)2} = {n,n(1(1,,2,,)1,,2)2}
1382FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)+ψΩM+1M-I(1,0)(0)))
ψ(ψχM(1,0)(0)(0)+ψΩM+1χM(1,0)(0)(0)))
ψ(ψχχ1M(2;0)(0)(0)(0)+ψΩM+1χχ1M(2;0)(0)(0)(0)))
{n,n(1(1(1(1,,2,,)1(1(1,,2,,)1,,2)2)2,,)3)2}
1383FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)+ΩM+ω)
ψ(ψχM(1,0)(0)(0)+ΩM+ω)
ψ(ψχχ1M(2;0)(0)(0)(0)+ΩM+ω)
{n,n(1(1,,1,2(1,,2,,)2)2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1384FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)+Mω)
ψ(ψχM(1,0)(0)(0)+Mω)
{n,n(1(1(1,,2,,)1,2)2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1385FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)+MM)
ψ(ψχM(1,0)(0)(0)+MM)
ψ(ψχχ1M(2;0)(0)(0)(0)+MM)
{n,n(1(1(1,,2,,)1(1(1,,2,,)2)2)2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1386FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(0)×2)
ψ(ψχM(1,0)(0)(0)×2)
ψ(ψχχ1M(2;0)(0)(0)(0)×2)
{n,n(1(1(1,,2,,)1(1(1,,2,,)1,,2)2)2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1387FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,0)(0)+1) = ψ(χMψM-I(1,0)(0)+1(0))
ψ(1st uncountable after 1st fixed point of weakly Mahlo)
ψ(ΩψχM(1,0)(0)(0)+1) = ψ(χMψχM(1,0)(0)(0)+1(0))
ψ(Ωψχχ1M(2;0)(0)(0)(0)+1) = ψ(χMψχχ1M(2;0)(0)(0)(0)+1(0))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1388FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,0)(0)+2) = ψ(χMψM-I(1,0)(0)+1(1))
ψ(2nd uncountable after 1st fixed point of weakly Mahlo)
ψ(ΩψχM(1,0)(0)(0)+2) = ψ(χMψχM(1,0)(0)(0)+1(1))
ψ(Ωψχχ1M(2;0)(0)(0)(0)+2) = ψ(χMψχχ1M(2;0)(0)(0)(0)+1(1))
{n,n(1,,3(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1389FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)+ΩΩΩΩ...) = ψ(χMM-I(1,0)ΩΩΩΩ...))
ψ((1st fixed point of uncountable) after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)+ΩΩΩΩ...) = ψ(χMχM(1,0)(0)+1ΩΩΩΩ...))
ψ(Ωχχ1M(2;0)(0)(0)+ΩΩΩΩ...) = ψ(χMχχ1M(2;0)(0)(0)+1ΩΩΩΩ...))
{n,n(1,,1(1,,1,,2)2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1390FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0)(0)+1) = ψ(χMψM-I(1,0)(0)+1MψM-I(1,0)(0)+1(...))) = ψ(ΩΩ...ΩψM-I(1,0)(0)+1)
ψ(1st weakly Mahlo after 1st fixed point of weakly Mahlo) or ψ(1st fixed point of (uncountable after 1st fixed point of weakly Mahlo))
ψ(MψχM(1,0)(0)(0)+1) = ψ(χMψχM(1,0)(0)(0)+1MψχM(1,0)(0)(0)+1(...))) = ψ(ΩΩ...ΩψχM(1,0)(0)(0)+1)
ψ(Mψχχ1M(2;0)(0)(0)(0)+1) = ψ(χMψχχ1M(2;0)(0)(0)(0)+1Mψχχ1M(2;0)(0)(0)(0)+1(...))) = ψ(ΩΩ...Ωψχχ1M(2;0)(0)(0)(0)+1)
{n,n(1,,1,,2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1391FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0)(0)+1MψI-M(1,0)(0)+1)
ψ(MψχM(1,0)(0)(0)+1MψχM(1,0)(0)(0)+1)
ψ(Mψχχ1M(2;0)(0)(0)(0)+1Mψχχ1M(2;0)(0)(0)(0)+1)
{n,n(1(1,,2,,)2(1(1,,2,,)1,,2)2)2}
1392FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩMψM-I(1,0)(0)+1+1) = ψ(MψM-I(1,0)(0)+1MψI-M(1,0)(0)+1...)
ψ(1st uncountable after 1st weakly Mahlo after 1st fixed point of weakly Mahlo)
ψ(ΩMψχM(1,0)(0)(0)+1+1) = ψ(MψχM(1,0)(0)(0)+1MψχM(1,0)(0)(0)+1...)
ψ(ΩMψχχ1M(2;0)(0)(0)(0)+1+1) = ψ(Mψχχ1M(2;0)(0)(0)(0)+1Mψχχ1M(2;0)(0)(0)(0)+1...)
{n,n(1,,2(1,,2,,)2(1(1,,2,,)1,,2)2)2}
1393FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0)(0)+2) = ψ(ΩΩ...MψM-I(1,0)(0)+1)
ψ(2nd weakly Mahlo after 1st fixed point of weakly Mahlo) or ψ(1st fixed point of uncountable after weakly Mahlo after 1st fixed point of weakly Mahlo)
ψ(MψχM(1,0)(0)(0)+2) = ψ(ΩΩ...MψχM(1,0)(0)(0)+1)
ψ(Mψχχ1M(2;0)(0)(0)(0)+2) = ψ(ΩΩ...Mψχχ1M(2;0)(0)(0)(0)+1)
{n,n(1,,1,,2(1,,2,,)2(1(1,,2,,)1,,2)2)2}
1394FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0)(0)+ω)
ψ(ω-th weakly Mahlo after 1st fixed point of weakly Mahlo)
ψ(MψχM(1,0)(0)(0)+ω)
ψ(Mψχχ1M(2;0)(0)(0)(0)+ω)
{n,n(1(1,,2,,)1,2(1(1,,2,,)1,,2)2)2}
1395FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0)(0)×2)
ψ(1st fixed point of weakly Mahlo-th weakly Mahlo after 1st fixed point of weakly Mahlo)
ψ(MψχM(1,0)(0)(0)×2)
ψ(Mψχχ1M(2;0)(0)(0)(0)×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)1,,2)2)2(1(1,,2,,)1,,2)2)2}
1396FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MMψM-I(1,0)(0)+ω)
ψ(ω-th weakly Mahlo after 1st fixed point of weakly Mahlo-th weakly Mahlo after 1st fixed point of weakly Mahlo)
ψ(MMψχM(1,0)(0)(0)+ω)
ψ(MMψχχ1M(2;0)(0)(0)(0)+ω)
{n,n(1(1,,2,,)1(1(1,,2,,)1,2(1(1,,2,,)1,,2)2)2(1(1,,2,,)1,,2)2)2}
1397FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(1)) = ψ(MM...MψM-I(1,0)(0)+1)
ψ(2nd fixed point of weakly Mahlo)
ψ(ψχM(1,0)(0)(1)) = ψ(MM...MψχM(1,0)(0)(0)+1)
ψ(ψχχ1M(2;0)(0)(0)(1)) = ψ(MM...Mψχχ1M(2;0)(0)(0)(0)+1)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)3)2}
1398FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(2)) = ψ(MM...MψM-I(1,0)(1)+1)
ψ(3d fixed point of weakly Mahlo)
ψ(ψχM(1,0)(0)(2)) = ψ(MM...MψχM(1,0)(0)(1)+1)
ψ(ψχχ1M(2;0)(0)(0)(2)) = ψ(MM...Mψχχ1M(2;0)(0)(0)(1)+1)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)4)2}
1399FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)(M))
ψ(Mahlo-th fixed point of weakly Mahlo)
ψ(ψχM(1,0)(0)(M))
ψ(ψχχ1M(2;0)(0)(0)(M))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)2)2)2}
1400FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0)M-I(1,0)(0)))
ψ(1st fixed point of weakly Mahlo-th fixed point of weakly Mahlo)
ψ(ψχM(1,0)(0)χM(1,0)(0)(0))
ψ(ψχχ1M(2;0)(0)(0)χχ1M(2;0)(0)(0)(0))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1(1(1,,2,,)1,,2)2)2)2}
1401FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0)) = ψ(ψM-I(1,0)M-I(1,0)M-I(1,0)(...)))) = ψ(M(0,α)-I(1,0))
ψ(1st weakly inaccessible limit of weakly Mahlo)
ψ(χM(1,0)(0)) = ψ(ψχM(1,0)(0)χM(1,0)(0)χM(1,0)(0)(...))))
ψ(χχ1M(2;0)(0)(0)) = ψ(ψχχ1M(2;0)(0)(0)χχ1M(2;0)(0)(0)χχ1M(2;0)(0)(0)(...))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)2)2}
1402FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0)×2)
ψ(χM(1,0)(0)×2)
ψ(χχ1M(2;0)(0)(0)×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)3)2}
1403FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0)2)
ψ(χM(1,0)(0)2)
ψ(χχ1M(2;0)(0)(0)×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)2)2}
1404FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0)M-I(1,0))
ψ(χM(1,0)(0)χM(1,0)(0))
ψ(χχ1M(2;0)(0)(0)χχ1M(2;0)(0)(0))
{n,n(1(1(1,,2,,)1,,2)2(1,,2,,)1,,2)2}
1405FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)+1) = ψ(χMM-I(1,0)(0))
ψ(1st uncountable after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)+1) = ψ(χMχM(1,0)(0)+1(1))
ψ(Ωχχ1M(2;0)(0)(0)+1) = ψ(χMχχ1M(2;0)(0)(0)+1(1))
{n,n(1,,2(1,,2,,)1,,2)2}
1406FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)+2) = ψ(χMM-I(1,0)(1))
ψ(2nd uncountable after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)+2) = ψ(χMχM(1,0)(0)+1(2))
ψ(Ωχχ1M(2;0)(0)(0)+2) = ψ(χMχχ1M(2;0)(0)(0)+1(2))
{n,n(1,,3(1,,2,,)1,,2)2}
1407FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)+ΩΩΩΩ...) = ψ(χMM-I(1,0)ΩΩΩΩ...))
ψ((1st fixed point of uncountable) after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)+ΩΩΩΩ...) = ψ(χMχM(1,0)(0)+1ΩΩΩΩ...))
ψ(Ωχχ1M(2;0)(0)(0)+ΩΩΩΩ...) = ψ(χMχχ1M(2;0)(0)(0)+1ΩΩΩΩ...))
{n,n(1,,1(1,,1,,2)2(1,,2,,)1,,2)2}
1408FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)+ψM-I(1,0)(0)) = ψ(χMM-I(1,0)+1I-M(1,0)(0)))
ψ(1st weakly inaccessible limit of weakly Mahlo-th uncountable after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)+ψχM(1,0)(0)(0)) = ψ(χMχM(1,0)(0)+1χM(1,0)(0)(0)))
ψ(Ωχχ1M(2;0)(0)(0)+ψχχ1M(2;0)(0)(0)(0)) = ψ(χMχχ1M(2;0)(0)(0)+1χχ1M(2;0)(0)(0)(0)))
{n,n(1,,1(1(1,,2,,)1(1(1,,2,,)1,,2)2)2(1,,2,,)1,,2)2}
1409FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0)×2) = ψ(χMM-I(1,0)+1MM-I(1,0)+1(0)))
ψ(1st uncountable after 1st weakly inaccessible limit of weakly Mahlo-th uncountable after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(0)×2) = ψ(χMχM(1,0)(0)+1MχM(1,0)(0)+1(0)))
ψ(Ωχχ1M(2;0)(0)(0)×2) = ψ(χMχχ1M(2;0)(0)(0)+1Mχχ1M(2;0)(0)(0)+1(0)))
{n,n(1,,1(1(1,,2,,)1,,2)2(1,,2,,)1,,2)2}
1410FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)+1) = ψ(ΩΩΩ...ΩM-I(1,0)+1)
ψ(1st weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo) or ψ(1st fixed point of (uncountable after 1st weakly inaccessible limit of weakly Mahlo))
ψ(MχM(1,0)(0)+1) = ψ(χMχM(1,0)(0)+1MχM(1,0)(0)+1(...)))
ψ(Mχχ1M(2;0)(0)(0)+1) = ψ(χMχχ1M(2;0)(0)(0)+1Mχχ1M(2;0)(0)(0)+1(...)))
{n,n(1,,1,,2(1,,2,,)1,,2)2}
1411FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)+1×2)
ψ(MχM(1,0)(0)+1×2)
ψ(Mχχ1M(2;0)(0)(0)+1×2)
{n,n(1,,1,,3(1,,2,,)1,,2)2}
1412FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)+1MM-I(1,0)+1)
ψ(MχM(1,0)(0)+1MχM(1,0)(0)+1)
ψ(Mχχ1M(2;0)(0)(0)+1Mχχ1M(2;0)(0)(0)+1)
{n,n(1(1,,2,,)2,,2)2}
1413FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩMM-I(1,0)+1+1)
ψ(1st uncountable after 1st weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩMχM(1,0)(0)+1+1)
ψ(ΩMχχ1M(2;0)(0)(0)+1+1)
{n,n(1,,2(1,,2,,)2,,2)2}
1414FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)+2)
ψ(2nd weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(MχM(1,0)(0)+2)
ψ(Mχχ1M(2;0)(0)(0)+2)
{n,n(1,,1,,2(1,,2,,)2,,2)2}
1415FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)+ω)
ψ(ω-th weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(MχM(1,0)(0)+ω)
ψ(Mχχ1M(2;0)(0)(0)+ω)
{n,n(1(1,,2,,)1,2,,2)2}
1416FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0)×2)
ψ(1st weakly inaccessible limit of weakly Mahlo-th weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(MχM(1,0)(0)×2)
ψ(Mχχ1M(2;0)(0)(0)×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)2,,2)2}
1417FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MMM-I(1,0)+1)
ψ(1st weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo-th weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(MMχM(1,0)(0)+1)
ψ(MMχχ1M(2;0)(0)(0)+1)
{n,n(1(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)1,,2)2,,2)2,,2)2}
1418FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,1)(0)) = ψ(MMMM...M-I(1,0)+1)
ψ(1st fixed point of weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(1)(1)) = ψ(MMM...MχM(1,0)(0)+1)
ψ(ψχχ1M(2;0)(0)(1)(1)) = ψ(MMM...Mχχ1M(2;0)(0)(0)+1)
{n,n(1(1,,2,,)1,,3)2}
1419FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,1)(0)+1) = ψ(χMψM-I(1,1)(0)+1(0))
ψ(1st uncountable after 1st fixed point of weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ΩψχM(1,0)(1)(1)+1) = ψ(χMψχM(1,0)(1)(1)+1(1))
ψ(Ωψχχ1M(2;0)(0)(1)(1)+1) = ψ(χMψχχ1M(2;0)(0)(1)(1)+1(1))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,2)2)2}
1420FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,1)(ω))
ψ(ω-th fixed point of weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(1)(ω))
ψ(ψχχ1M(2;0)(0)(1)(ω))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,3)1,2)2}
1421FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,1)) = ψ(ψM-I(1,1)M-I(1,1)(...)))
ψ(2nd weakly inaccessible limit of weakly Mahlo)
ψ(χM(1,0)(1)) = ψ(ψχM(1,0)(1)χM(1,0)(1)(...))
ψ(χχ1M(2;0)(0)(1)) = ψ(ψχχ1M(2;0)(0)(1)χχ1M(2;0)(0)(1)(...))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,3)1(1(1,,2,,)1,,3)2,,2)2}
1422FGH ψ-function
ψ(cardinal)

FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,1)+1) = ψ(χMM-I(1,1)(1))
ψ(1st uncountable after 2nd weakly inaccessible limit of weakly Mahlo)
ψ(ΩχM(1,0)(1)+1) = ψ(χMχM(1,0)(1)+1(2))
ψ(Ωχχ1M(2;0)(0)(1)+1) = ψ(χMχχ1M(2;0)(0)(1)+1(2))
{n,n(1,,2(1,,2,,)1,,3)2}
1423FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,1)+1) = ψ(ΩΩΩ...ΩM-I(1,1)+1)
ψ(1st weakly Mahlo after 2nd weakly inaccessible limit of weakly Mahlo) or ψ(1st fixed point of (uncountable after 2nd weakly inaccessible limit of weakly Mahlo))
ψ(MχM(1,0)(1)+1) = ψ(χMχM(1,0)(1)+1MχM(1,0)(1)+1(...)))
ψ(Mχχ1M(2;0)(0)(1)+1) = ψ(χMχχ1M(2;0)(0)(1)+1Mχχ1M(2;0)(0)(1)+1(...)))
{n,n(1,,1,,2(1,,2,,)1,,3)2}
1424FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,1)+ω)
ψ(ω-th weakly Mahlo after 2nd weakly inaccessible limit of weakly Mahlo)
ψ(MχM(1,0)(1)+ω)
ψ(Mχχ1M(2;0)(0)(1)+ω)
{n,n(1(1,,2,,)1,2,,3)2}
1425FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,2)(0)) = ψ(MMMM...M-I(1,1)+1)
ψ(1st fixed point of weakly Mahlo after 2nd weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(2)(2)) = ψ(MMM...MχM(1,0)(1)+1)
ψ(ψχχ1M(2;0)(0)(2)(2)) = ψ(MMM...Mχχ1M(2;0)(0)(1)+1)
{n,n(1(1,,2,,)1,,4)2}
1426FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,2))
ψ(3d weakly inaccessible limit of weakly Mahlo)
ψ(χM(1,0)(2))
ψ(χχ1M(2;0)(0)(2))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,4)1(1(1,,2,,)1,,4)2,,3)2}
1427FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)(0))
ψ(1st fixed point of weakly Mahlo after sup(n)-th weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(ω)(ω))
ψ(ψχχ1M(2;0)(0)(ω)(ω))
{n,n(1(1,,2,,)1,,1,2)2}
1428FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)(0)×2)
ψ(ψχM(1,0)(ω)(ω)×2)
ψ(ψχχ1M(2;0)(0)(ω)(ω)×2)
{n,n(1(1(1,,2,,)1,,1,2)2(1,,2,,)1,,1,2)2}
1429FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,ω)(0)+1) = ψ(χMψM-I(1,ω)(0)+1(0))
ψ(1st uncountable after 1st fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(ΩψχM(1,0))(ω)+1) = ψ(χMψχM(1,0)(ω)(ω)+1(ω))
ψ(Ωψχχ1M(2;0)(0))(ω)+1) = ψ(χMψχχ1M(2;0)(0)(ω)(ω)+1(ω))
{n,n(1,,2(1,,2,,)1,,1,2)2}
1430FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,ω)(0)+2) = ψ(χMψM-I(1,ω)(0)+1(1))
ψ(2nd uncountable after 1st fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(ΩψχM(1,0)(ω)(ω)+2) = ψ(χMψχM(1,0)(ω)(ω)+1(ω+1)
ψ(Ωψχχ1M(2;0)(0)(ω)(ω)+2) = ψ(χMψχχ1M(2;0)(0)(ω)(ω)+1(ω+1)
{n,n(1,,3(1,,2,,)1,,1,2)2}
1431FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,ω)(0)+1)
ψ(1st weakly Mahlo after 1st fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(MψχM(1,0))(ω)+1
ψ(Mψχχ1M(2;0)(0))(ω)+1
{n,n(1,,1,,2(1,,2,,)1,,1,2)2}
1432FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,ω)(0)+ω)
ψ(ω-th weakly Mahlo after 1st fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(MψχM(1,0))(ω)+ω)
ψ(Mψχχ1M(2;0)(0))(ω)+ω)
{n,n(1(1,,2,,)1,2,,1,2)2}
1433FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)(1)) = ψ(MM...MψM-I(1,ω)(0)+1)
ψ(2nd fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(ω)(ω+1)) = ψ(MM...MψχM(1,0)(ω)(ω)+1)
ψ(ψχχ1M(2;0)(0)(ω)(ω+1)) = ψ(MM...Mψχχ1M(2;0)(0)(ω)(ω)+1)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2)2,,1,2)2} = {n,n(1(1,,2,,)1,,2,2)2}
1434FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)(2)) = ψ(MM...MψM-I(1,ω)(1)+1)
ψ(3d fixed point of weakly Mahlo after n-th weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)(ω)(ω+2)) = ψ(MM...MψχM(1,0)(ω)(ω+1)+1)
ψ(ψχχ1M(2;0)(0)(ω)(ω+2)) = ψ(MM...Mψχχ1M(2;0)(0)(ω)(ω+1)+1)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2)3,,1,2)2}
1435FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)(ω))
ψ(ψχM(1,0)(ω)(ω×2))
ψ(ψχχ1M(2;0)(0)(ω)(ω×2))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2)1,2,,1,2)2}
1436FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω)ω))
ψ(ψχM(1,0)(ω)ω))
ψ(ψχχ1M(2;0)(0)(ω)ω))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2)1,1,2,,1,2)2}
1437FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ω)) = ψ(ψM-I(1,ω)M-I(1,ω)(...)))
ψ(ω-th weakly inaccessible limit of weakly Mahlo)
ψ(χM(1,0)(ω)) = ψ(ψχM(1,0)(ω)χM(1,0)(ω)(...)))
ψ(χχ1M(2;0)(0)(ω)) = ψ(ψχχ1M(2;0)(0)(ω)χχ1M(2;0)(0)(ω)(...)))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2)1(1(1,,2,,)1,,2,2)2,,1,2)2}
1438FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω+1)(0))
ψ(χM(1,0)(ω+1))
ψ(χχ1M(2;0)(0)(ω+1))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,3,2)1(1(1,,2,,)1,,3,2)2,,2,2)2}
1439FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ω×2)(0))
ψ(χM(1,0)(ω×2))
ψ(χχ1M(2;0)(0)(ω×2))
{n,n(1(1,,2,,)1,,1,3)2}
1440FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M-I(1,0))(0))
ψ(1st fixed point of weakly Mahlo after α-fixed point of weakly Mahlo-th weakly inaccessible limit of weakly Mahlo)
ψ(ψχM(1,0)M(1,0)(0))M(1,0)(0)))
ψ(ψχχ1M(2;0)(0)χ1M(2;0)(0)(0))χ1M(2;0)(0)(0)))
{n,n(1(1,,2,,)1,,1(1(1,,2,,)1,,2)2)2}
1441FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M-I(1,0))(M-I(1,0)×ω))
ψ(ψχM(1,0)M(1,0)(0))M(1,0)(0)×ω))
ψ(ψχχ1M(2;0)(0)χ1M(2;0)(0)(0))χ1M(2;0)(0)(0)×ω))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)2,,1(1(1,,2,,)1,,2)1,2)2}
1442FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(1,0))) = ψ(ψM-I(1,M-I(1,0))M-I(1,M-I(1,0))(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo-th weakly inaccessible limit of weakly Mahlo)
ψ(χM(1,0)M(1,0)(0))) = ψ(ψχM(1,0)M(1,0)(0))M(1,0)χM(1,0)M(1,0)(0))M(1,0)(...)))))
ψ(χχ1M(2;0)(0)χ1M(2;0)(0)(0))) = ψ(ψχχ1M(2;0)(0)χ1M(2;0)(0)(0))χ1M(2;0)(0)χχ1M(2;0)(0)χ1M(2;0)(0)(0))χ1M(2;0)(0)(...)))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1(1,,2,,)1,,2)2)1(1(1,,2,,)1,,2(1(1,,2,,)1,,2)2)2,,1(1(1,,2,,)1,,2)2)2}
1443FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
ψ(ψM(1,0)(cardinal))

SAN
ψ(ψM-I(2,0)(0)) = ψ(M-I(1,M-I(1,M-I(1,...))))
ψ(1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)) = ψ(χM(1,0)M(1,0)M(1,0)(...))))
ψ(χ1M(2;0)(0)) = ψ(χχ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)(...))))
ψ(1st weakly Mahlo limit of weakly Mahlo)
{n,n(1(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2} = {n,n(1(1,,2,,)1,,1,,2)2}
1444FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)+Mω)
ψ(M(1,0)+Mω)
ψ(χ1M(2;0)(0)+Mω)
{n,n(1(1(1,,2,,)1,2)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1445FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)+ψM-I(1,0)M-I(2,0)(0)))
ψ(M(1,0)+ψχM(1,0)(0)(M(1,0)))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)(0)1M(2;0)(0)))
{n,n(1(1(1,,2,,)1(1(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1446FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)+M-I(1,0))
ψ(M(1,0)+χM(1,0)(0))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)(0))
{n,n(1(1(1,,2,,)1,,2)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1447FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)+MM-I(1,0)+ω)
ψ(M(1,0)+MχM(1,0)(0)+ω)
ψ(χ1M(2;0)(0)+Mχχ1M(2;0)(0)(0)+ω)
{n,n(1(1(1,,2,,)1,2,,2)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1448FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)+M-I(1,1))
ψ(M(1,0)+χM(1,0)(1))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)(1))
{n,n(1(1(1,,2,,)1,,3)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1449FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(0)×2) = ψ(ψM-I(2,0)(0)+ψM-I(2,0)(0))
ψ(M(1,0)+ψχM(1,0)(M(1,0))(M(1,0)))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0)))
{n,n(1(1(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1450FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(2,0)(0)+1) = ψ(χMψM-I(2,0)(0)+1(0))
ψ(1st uncountable after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ΩψχM(1,0)(M(1,0))(M(1,0))+1) = ψ(M(1,0)+χMψχM(1,0)(M(1,0))(M(1,0))+1(M(1,0)))
ψ(χ1M(2;0)(0)+Ωψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1) = ψ(χ1M(2;0)(0)+χMψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+11M(2;0)(0)))
{n,n(1,,2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1451FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(2,0)(0)+1) = ψ(χMψM-I(2,0)(0)+1MψM-I(2,0)(0)+1(...)))
ψ(1st weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+MψχM(1,0)(M(1,0))(M(1,0))+1)
ψ(χ1M(2;0)(0)+Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)
{n,n(1,,1,,2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2}
1452FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(2,0)(0)+1MψM-I(2,0)(0)+1)
ψ(M(1,0)+MψχM(1,0)(M(1,0))(M(1,0))+1MψχM(1,0)(M(1,0))(M(1,0))+1)
ψ(χ1M(2;0)(0)+Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)
{n,n(1(1,,2,,)2,,1(1(1,,2,,)1,,1,,2)2)2}
1453FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(2,0)(0)+ω)
ψ(ω-th weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+MψχM(1,0)(M(1,0))(M(1,0))+ω)
ψ(χ1M(2;0)(0)+Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+ω)
{n,n(1(1,,2,,)1,2,,1(1(1,,2,,)1,,1,,2)2)2}
1454FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ψM-I(2,0)(0)+1)(0)) = ψ(MM...MψM-I(2,0)(0)+1)
ψ(1st fixed point of weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+1)χM(1,0)(M(1,0)(M(1,0))+1)) = ψ(MM...MψχM(1,0)(M(1,0))(M(1,0))+1)
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))+1)) = ψ(MM...Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)
{n,n(1(1,,2,,)1,,2(1(1,,2,,)1,,1,,2)2)2}
1455FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ψM-I(2,0)(0)+1)) = ψ(ψM-I(1,ψM-I(2,0)(0)+1)M-I(1,ψM-I(2,0)(0)+1)M-I(1,ψM-I(2,0)(0)+1)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+χM(1,0)χM(1,0)(M(1,0))(M(1,0))+1)) = ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+1)χM(1,0)(M(1,0)(M(1,0))+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+1)χM(1,0)(M(1,0)(M(1,0))+...)))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)) = ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))+...))
{n,n(1,,2(1,,2,,)1,,2(1(1,,2,,)1,,1,,2)2)2}
1456FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ψM-I(2,0)(0)+1)(0)) = ψ(MMM...M-I(1,ψM-I(2,0)(0)+1))
ψ(1st fixed point of weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+2)χM(1,0)(M(1,0)(M(1,0))+2)) = ψ(MM...MψχM(1,0)(M(1,0))(M(1,0))+1)
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+2)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))+2)) = ψ(MM...Mψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+1)
{n,n(1(1,,2,,)1,,3(1(1,,2,,)1,,1,,2)2)2}
1457FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ψM-I(2,0)(0)+2)) = ψ(ψM-I(1,ψM-I(2,0)(0)+2)M-I(1,ψM-I(2,0)(0)+2)M-I(1,ψM-I(2,0)(0)+2)(...)))
ψ(2nd weakly inaccessible limit of weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+χM(1,0)χM(1,0)(M(1,0))(M(1,0))+2)) = ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+2)χM(1,0)(M(1,0)(M(1,0))+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+2)χM(1,0)(M(1,0)(M(1,0))+...)))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+2)) = ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+2)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))+...))
{n,n(1,,2(1,,2,,)1,,3(1(1,,2,,)1,,1,,2)2)2}
1458FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ψM-I(2,0)(0)+ω))
ψ(ω-th weakly inaccessible limit of weakly Mahlo after 1st fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))+ω)χM(1,0)(M(1,0)(M(1,0))+ω))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))+ω)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))+ω))
{n,n(1(1,,2,,)1,,1,2(1(1,,2,,)1,,1,,2)2)2}
1459FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ψM-I(2,0)(0)×2))
ψ(M(1,0)+ψχM(1,0)χM(1,0)(M(1,0))(M(1,0))×2)χM(1,0)(M(1,0)(M(1,0))×2))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0))×2)χχ1M(2;0)(0)1M(2;0)(0)1M(2;0)(0))×2))
{n,n(1(1,,2,,)1,,1(1(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2)2(1(1,,2,,)1,,1,,2)2)2}
1460FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0)(1)) = ψ(M-I(1,M-I(1,...M-I(1,ψM-I(2,0)(0)+1)...)))
ψ(2nd fixed point of weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ψχM(1,0)(M(1,0))(M(1,0)+1))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)1M(2;0)(0))1M(2;0)(0)+1))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,1,,2)3)2}
1461FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(2,0)) = ψ(ψM-I(2,0)M-I(2,0)M-I(2,0)(...))))
ψ(1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+χM(1,0)(M(1,0)))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,1,,2)1(1(1,,2,,)1,,1,,2)2)2}
1462FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(2,0)+1) = ψ(χMM-I(2,0)+1(0))
ψ(1st uncountable after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ΩχM(1,0)(M(1,0))+1) = ψ(M(1,0)+χMχM(1,0)(M(1,0))+1(M(1,0)+1))
ψ(χ1M(2;0)(0)+Ωχχ1M(2;0)(0)1M(2;0)(0))+1) = ψ(χ1M(2;0)(0)+χMχχ1M(2;0)(0)1M(2;0)(0))+11M(2;0)(0)+1))
{n,n(1,,2(1,,2,,)1,,1,,2)2}
1463FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(2,0)+1) = ψ(ΩΩΩ...ΩM-I(2,0)+1)
ψ(1st weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo) or ψ(1st fixed point of (uncountable after 1st 2-weakly inaccessible limit of weakly Mahlo))
ψ(M(1,0)+MχM(1,0)(M(1,0))+1) = ψ(M(1,0)+χMχM(1,0)(M(1,0))+1(M(1,0)+χMχM(1,0)(M(1,0))+1(M(1,0)+...)))
ψ(χ1M(2;0)(0)+Mχχ1M(2;0)(0)1M(2;0)(0))+1) = ψ(χ1M(2;0)(0)+χMχχ1M(2;0)(0)1M(2;0)(0))+11M(2;0)(0)+χMχχ1M(2;0)(0)1M(2;0)(0))+11M(2;0)(0)+...)))
{n,n(1,,1,,2(1,,2,,)1,,1,,2)2}
1464FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(2,0)+ω)
ψ(ω-th weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+MχM(1,0)(M(1,0))+ω)
ψ(χ1M(2;0)(0)+Mχχ1M(2;0)(0)1M(2;0)(0))+ω)
{n,n(1(1,,2,,)1,2,,1,,2)2}
1465FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(2,0)×2)
ψ(1st 2-weakly inaccessible limit of weakly Mahlo-th weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+MχM(1,0)(M(1,0))×2)
ψ(χ1M(2;0)(0)+Mχχ1M(2;0)(0)1M(2;0)(0))×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,1,,2)2,,1,,2)2}
1466FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M-I(2,0)+1)(0)) = ψ(MMMM...M-I(2,0)+1)
ψ(1st fixed point of weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+ψχM(1,0)(M(1,0)+1)(M(1,0)+1))
ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)1M(2;0)(0)+1)1M(2;0)(0)+1))
{n,n(1(1,,2,,)1,,2,,2)2}
1467FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(2,0)+1)) = ψ(ψM-I(1,M-I(2,0)+1)M-I(1,M-I(2,0)+1)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+χM(1,0)(M(1,0)+1)) = ψ(M(1,0)+ψχM(1,0)(M(1,0)+1)(M(1,0)+ψχM(1,0)(M(1,0)+1)(M(1,0)+...)))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)+1)) = ψ(χ1M(2;0)(0)+ψχχ1M(2;0)(0)1M(2;0)(0)+1)1M(2;0)(0)+ψχχ1M(2;0)(0)1M(2;0)(0)+1)1M(2;0)(0)+...)))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,,2)1(1(1,,2,,)1,,2,,2)2,,1,,2)2}
1468FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(2,0)+ω))
ψ(ω-th weakly inaccessible limit of weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)+χM(1,0)(M(1,0)+ω))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)+ω))
{n,n(1(1,,2,,)1,,1,2,,2)2}
1469FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(2,0)×2))
ψ(M(1,0)+χM(1,0)(M(1,0)+χM(1,0)(M(1,0))))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0))))
{n,n(1,,2(1,,2,,)1,,1(1(1,,2,,)1,,1,,2)2,,2)2}
1470FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(1,M-I(2,0)+1))))
ψ(M(1,0)+χM(1,0)(M(1,0)+χM(1,0)(M(1,0)+1)))
ψ(χ1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)+χχ1M(2;0)(0)1M(2;0)(0)+1)))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,1(1(1,,2,,)1,,2,,2)2,,2)1(1(1,,2,,)1,,1(1(1,,2,,)1,,2,,2)2,,2)2,,1,,2)2}
1471FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,1)(0)) = ψ(M-I(1,M-I(1,...M-I(2,0)+1)...)))
ψ(1st fixed point of weakly inaccessible limit of weakly Mahlo after 1st 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)×2)
ψ(χ1M(2;0)(0)×2)
{n,n(1(1,,2,,)1,,1,,3)2}
1472FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(2,1)) = ψ(ψM-I(2,1)M-I(2,1)M-I(2,1)(...))))))
ψ(2nd 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)×2+χM(1,0)(M(1,0)×2))
ψ(χ1M(2;0)(0)×2+χχ1M(2;0)(0)1M(2;0)(0)×2))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,1,,3)1(1(1,,2,,)1,,1,,3)2)2}
1473FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,2)(0)) = ψ(M-I(1,M-I(1,...M-I(2,1)+1)...)))
ψ(1st fixed point of weakly inaccessible limit of weakly Mahlo after 2nd 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)×3)
ψ(χ1M(2;0)(0)×3)
{n,n(1(1,,2,,)1,,1,,4)2}
1474FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(2,2)) = ψ(ψM-I(2,2)M-I(2,2)M-I(2,2)(...))))))
ψ(3d 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)×3+χM(1,0)(M(1,0)×3))
ψ(χ1M(2;0)(0)×3+χχ1M(2;0)(0)1M(2;0)(0)×3))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,1,,4)1(1(1,,2,,)1,,1,,4)2)2}
1475FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(3,0)(0)) = ψ(M-I(2,M-I(2,M-I(2,...))))
ψ(1st fixed point of 2-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)2)
ψ(χ1M(2;0)(0)2)
{n,n(1(1,,2,,)1,,1,,1,,2)2}
1476FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(3,0)) = ψ(ψM-I(3,0)M-I(3,0)M-I(3,0)(...))))))
ψ(3-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)2M(1,0)(M(1,0)2))
ψ(χ1M(2;0)(0)2χ1M(2;0)(0)1M(2;0)(0)2))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,1,,1,,2)1(1(1,,2,,)1,,1,,1,,2)2)2}
1477FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(ω,0)(0)) = ψ(M-I(sup(n),0))
ψ(1st fixed point of n-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)ω)
ψ(χ1M(2;0)(0)ω)
{n,n(1(1,,2,,)1(2,,)2)2}
1478FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(ω,0)) = ψ(ψM-I(ω,0)M-I(ω,0)M-I(ω,0)(...))))))
ψ(ω-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)ωM(1,0)(M(1,0)ω))
ψ(χ1M(2;0)(0)ωχ1M(2;0)(0)1M(2;0)(0)ω))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1(2,,)2)1(1(1,,2,,)1(2,,)2)2)2}
1479FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(M-I(ω,0),0))
ψ(ω-weakly inaccessible limit of weakly Mahlo-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)χM(1,0)(M(1,0)ω)M(1,0)(M(1,0)χM(1,0)(M(1,0)ω)))
ψ(χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)ω)χ1M(2;0)(0)1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)ω)))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)1(2,,)2)2)1(1(1,,2,,)1(1(1,,2,,)1(2,,)2)2)2)2}
1480FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0)(0)) = ψ(M-I(M-I(M-I(...,0),0),0))
ψ(1st fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)) = ψ(M(1,0)χM(1,0)(M(1,0)χM(1,0)(M(1,0)...))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)) = ψ(χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)...))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2} = {n,n(1(1,,2,,)1(1,,2,,)2)2}
1481FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0)(0)×2)
ψ(M(1,0)M(1,0)χM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0)))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0)))
{n,n(1(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1482FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩψM-I(1,0,0)(0)+1) = ψ(χMψM-I(1,0,0)(0)+1(0))
ψ(1st uncountable after 1st fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)MψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))+1(M(1,0)M(1,0)))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)Mψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))+11M(2;0)(0)χ1M(2;0)(0)))
{n,n(1,,2(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1483FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0,0)(0)+1) = ψ(χMψM-I(1,0,0)(0)+1MψM-I(1,0,0)(0)+1(...)))
ψ(1st weakly Mahlo after 1st fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+MψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))+1)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)+Mψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))+1)
{n,n(1,,1,,2(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1484FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MψM-I(1,0,0)(0)+ω)
ψ(ω-th weakly Mahlo after 1st fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+MψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))+ω)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)+Mψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))+ω)
{n,n(1(1,,2,,)1,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1485FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(ψM-I(1,0,0)(0),1)(0))
ψ(M(1,0)M(1,0)χM(1,0)(M(1,0)ψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0)))(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)ψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0)))1M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1486FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(ψM-I(1,0,0)(0),1))
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)ψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)ψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)1(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1487FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(ψM-I(1,0,0)(0),1)+ω)
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)ψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))+ω))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)ψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))+ω))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)1(1(1,,2,,)1,,2,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2,,1,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1488FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(ω,M-I(ψM-I(1,0,0)(0),1)+1))
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)ψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))+M(1,0)ω))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)ψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)ω))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(2,,)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)1(1(1,,2,,)1,,2(2,,)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2(2,,)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1489FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(ψM-I(1,0,0)(0),2))
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)ψχM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))×2))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)ψχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))×2))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)1(1(1,,2,,)1,,2(2,,)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2(2,,)2(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1490FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0)(1)) = ψ(M-I(M-I(M-I(M-I(...M-I(ψM-I(1,0,0)(0),1)...,0),0),0),0))
ψ(2nd fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)χM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)3)2}
1491FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0)M-I(1,0,0)(0)))
ψ(1st fixed point α-weakly inaccessible limit of weakly Mahlo-th fixed point α-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)χM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0)χM(1,0)(M(1,0)M(1,0))(M(1,0)M(1,0))))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))1M(2;0)(0)χ1M(2;0)(0))))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2)2}
1492FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,0)) = ψ(ψM-I(1,0,0)M-I(1,0,0)M-I(1,0,0)M-I(1,0,0)(...)))))
ψ(1st hyper-weakly inaccessible limit of weakly Mahlo or 1,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0)))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1493FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ΩM-I(1,0,0)+1) = ψ(χMM-I(1,0,0)+1(0))
ψ(1st uncountable after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)MχM(1,0)(M(1,0)M(1,0))+1(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)Mχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))+11M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1,,2(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1494FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0,0)+1) = ψ(χMM-I(1,0,0)+1MM-I(1,0,0)+1(...)))
ψ(1st weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+MχM(1,0)(M(1,0)M(1,0))+1)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)+Mχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))+1)
{n,n(1,,1,,2(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1495FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(MM-I(1,0,0)+ω)
ψ(1st weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+MχM(1,0)(M(1,0)M(1,0))+ω)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)+Mχχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))+ω)
{n,n(1(1,,2,,)1,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1496FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M-I(1,0,0)+1)(0)) = ψ(MMMM...M-I(1,0,0)+1)
ψ(1st fixed point of weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)χM(1,0)(M(1,0)M(1,0)+1)(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)+1)1M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1497FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(1,0,0)+1)) = ψ(ψM-I(1,M-I(1,0,0)+1)M-I(1,M-I(1,0,0)+1)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)1(1(1,,2,,)1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1498FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(1,0,0)+ω))
ψ(ω-th weakly inaccessible limit of weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0)+ω))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)+ω))
{n,n(1(1,,2,,)1,,1,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1499FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M-I(1,0,0)×2))
ψ(M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0))))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0))))
{n,n(1(1,,2,,)1,,1(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1500FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,M-I(1,0,0)+1)(0)) = ψ(M-I(1,(M-I(1,(M-I(1,...M-I(1,0,0)+1...))))))
ψ(1st fixed point of weakly inaccessible limit of weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+M(1,0))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0))
{n,n(1(1,,2,,)1,,1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1501FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(3,M-I(1,0,0)+1)(0)) = ψ(M-I(2,(M-I(2,(M-I(2,...M-I(1,0,0)+1...))))))
ψ(1st fixed point of 2-weakly inaccessible limit of weakly Mahlo after 1st hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+M(1,0)2)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)2)
{n,n(1(1,,2,,)1,,1,,1,,2(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1502FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,1)(0)) = ψ(M-I(M-I(M-I(M-I(...M-I(ψM-I(ψM-I(...,0)(0),0)(0),M-I(1,0,0)+1)...,0),0),0),0))
ψ(M(1,0)M(1,0)×2)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×2)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2,,2)2}
1503FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,1)) = ψ(ψM-I(1,0,1)M-I(1,0,1)M-I(1,0,1)(...))))
ψ(2nd hyper-weakly inaccessible limit of weakly Mahlo or 1,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)×2+χM(1,0)(M(1,0)M(1,0)×2))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×2+χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)×2))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,3)2}
1504FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,1,0)(0)) = ψ(M-I(1,0,M-I(1,0,M-I(1,0,...))))
ψ(1st fixed point of hyper-weakly inaccessible limit of weakly Mahlo or 1,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)+1)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)+1)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,1,,2)2}
1505FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,1,0)) = ψ(ψM-I(1,1,0)M-I(1,1,0)M-I(1,1,0)(...))))
ψ(2-hyper-weakly inaccessible limit of weakly Mahlo or 1,1-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)×2+χM(1,0)(M(1,0)M(1,0)+1))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×2+χχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)+1))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,1,,1(1,,1,,2)2)2}
1506FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,0,0)(0)) = ψ(M-I(1,M-I(1,M-I(1,...,0),0),0))
ψ(1st fixed point of α-hyper-weakly inaccessible limit of weakly Mahlo or 1,α-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)×2)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×2)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1507FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(2,0,0)) = ψ(ψM-I(2,0,0)M-I(2,0,0)M-I(2,0,0)(...))))
ψ(hyper-hyper-weakly inaccessible limit of weakly Mahlo or 2,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)×2M(1,0)(M(1,0)M(1,0)×2))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×2χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)×2))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1508FGH ψ-function
FGH ψ(ψM(1,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(ω,0,0)(0)) = ψ(M-I(sup(n),0,0))
ψ(M(1,0)M(1,0)×ω)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×ω)
{n,n(1(1,,2,,)1(2(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1509FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(ω,0,0)) = ψ(ψM-I(ω,0,0)M-I(ω,0,0)M-I(ω,0,0)(...))))
ψ(hyperω-weakly inaccessible limit of weakly Mahlo or ω,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)×ωM(1,0)(M(1,0)M(1,0)×ω))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)×ωχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)×ω))
{n,n(1(1,,2,,)1(2(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1510FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0,0)(0)) = ψ(M-I(M-I(M-I(...,0,0),0,0),0,0))
ψ(1st fixed point α,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)2)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)2)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)3,,)2)2}
1511FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,0,0)) = ψ(ψM-I(1,0,0,0)M-I(1,0,0,0)M-I(1,0,0,0)(...))))
ψ(1,0,0-hyper-weakly inaccessible limit of weakly Mahlo)
ψ(M(1,0)M(1,0)2M(1,0)(M(1,0)M(1,0)2))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)2χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)2))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)3,,)1,,2)2}
1512FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0,0,0)(0)) = ψ(M-I(M-I(M-I(...,0,0,0),0,0,0),0,0,0))
ψ(M(1,0)M(1,0)3)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)3)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)4,,)2)2}
1513FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,0,0,0)) = ψ(ψM-I(1,0,0,0,0)M-I(1,0,0,0,0)M-I(1,0,0,0,0)(...))))
ψ(M(1,0)M(1,0)3M(1,0)(M(1,0)M(1,0)3))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)3χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)3))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)4,,)1,,2)2}
1514FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,0,0,0,...)(0)) = ψ(ψM-I(1ω)(0))
ψ(M(1,0)M(1,0)ω)
ψ(χ1M(2;0)(0)χ1M(2;0)(0)ω)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)1,2,,)2)2}
1515FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,0,0,0,...)) = ψ(ψM-I(1,0,0,0,0,...)M-I(1,0,0,0,0,...)M-I(1,0,0,0,0,...)(...)))) = ψ(M-I(1ω))
ψ(M(1,0)M(1,0)ωM(1,0)(M(1,0)M(1,0)ω))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)ωχ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)ω))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)1,2,,)1,,2)2}
1516FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1M-I(1M-I(1M-I(1...))))(0))
ψ(M(1,0)M(1,0)M(1,0))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)1(1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1517FGH ψ-function
FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M-I(1M-I(1M-I(1M-I(1...)))))
ψ(M(1,0)M(1,0)M(1,0)M(1,0)(M(1,0)M(1,0)M(1,0)))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)2)1(1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1518FGH ψ(ψM(1,0))
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0)M(1,0)M(1,0)M(1,0))
ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0))
{n,n(1(1(1,,2,,)1(1,,2,,)2)2(1,,2,,)1(1,,2,,)2)2}
1519FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ΩM(1,0)+1) = ψ(χMM(1,0)+1(0)) = ψ(εM(1,0)+1) = ψ(M(1,0)M(1,0)M(1,0)M(1,0)...)
ψ(1st uncountable after 1st weakly Mahlo limit of weakly Mahlo)
ψ(Ωχ1M(2;0)(0)+1) = ψ(χMχ1M(2;0)(0)+1(0)) = ψ(εχ1M(2;0)(0)+1) = ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)...)
{n,n(1,,2(1,,2,,)1(1,,2,,)2)2}
1520FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(1,0)+1)
ψ(1st weakly Mahlo after 1st weakly Mahlo limit of weakly Mahlo)
ψ(Mχ1M(2;0)(0)+1)
{n,n(1,,1,,2(1,,2,,)1(1,,2,,)2)2}
1521FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(1,0)+ω)
ψ(ω-th weakly Mahlo after 1st weakly Mahlo limit of weakly Mahlo)
ψ(Mχ1M(2;0)(0)+ω)
{n,n(1(1,,2,,)1,2(1,,2,,)2)2}
1522FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M(1,0)+1)(0)) = ψ(MMM...M(1,0)+1.) = ψ(M(0,M(0,M(0,M(0,...M(1,0)+1)))))
ψ(1st fixed point of weakly Mahlo after 1st weakly Mahlo limit of weakly Mahlo)
ψ(ψχM(1,1)(0)(0)) = ψ(MMM...M(1,0)+1.)
ψ(ψχχ1M(2;0)(1)(1)(1)) = ψ(MMM...χ1M(2;0)(0)+1.)
{n,n(1(1,,2,,)1,,2(1,,2,,)2)2}
1523FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M(1,0)+1)(1)) = ψ(MM...MψM-I(1,M(1,0)+1)(0)+1)
ψ(2nd fixed point of weakly Mahlo after 1st weakly Mahlo limit of weakly Mahlo)
ψ(ψχM(1,1)(0)(1)) = ψ(MM...MψχM(1,1)(0)(0)+1)
ψ(ψχχ1M(2;0)(1)(1)(2)) = ψ(MM...Mψχχ1M(2;0)(1)(1)(1)+1)
{n,n(1(1,,2,,)1,,2(1,,2,,)2(1(1,,2,,)1,,2)3)2}
1524FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M(1,0)+1)) = ψ(ψM-I(1,M(1,0)+1)M-I(1,M(1,0)+1)(...)))
ψ(1st weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(χM(1,1)(0)) = ψ(ψχM(1,1)(0)χM(1,1)(0)χM(1,1)(0)(...))))
ψ(χχ1M(2;0)(1)(1)) = ψ(ψχχ1M(2;0)(1)(1)χχ1M(2;0)(1)(1)χχ1M(2;0)(1)(1)(...))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1,,2,,)2)1(1(1,,2,,)1,,2(1,,2,,)2)2(1,,2,,)2)2}
1525FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,M(1,1)+2)(0)) = ψ(MMM...M-I(1,M(1,0)+1)+1.)
ψ(1st fixed point of weakly Mahlo after 1st weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(ψχM(1,1)(1)(1)) = ψ(MMM...MχM(1,1)(0)+1)
ψ(ψχχ1M(2;0)(1)(2)(2)) = ψ(MMM...Mχχ1M(2;0)(1)(1)+1)
{n,n(1(1,,2,,)1,,3(1,,2,,)2)2}
1526FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M(1,0)+2)) = ψ(ψM-I(1,M(1,1)+2)M-I(1,M(1,1)+2)(...)))
ψ(2nd weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(χM(1,1)(1)) = ψ(ψχM(1,1)(1)χM(1,1)(1)χM(1,1)(1)(...))))
ψ(χχ1M(2;0)(1)(2)) = ψ(ψχχ1M(2;0)(1)(2)χχ1M(2;0)(1)(2)χχ1M(2;0)(1)(2)(...))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1,,2,,)2)1(1(1,,2,,)1,,2(1,,2,,)2)2(1,,2,,)2)2}
1527FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))
ψ(ψM(1,1)(cardinal))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,M(1,0)+1)(0)) = ψ(M-I(1,M-I(1,...M-I(1,M(1,0)+1)...)))
ψ(1st fixed point of weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)) = ψ(χM(1,1)M(1,1)M(1,1)(...))))
ψ(2nd weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)(1)) = ψ(χχ1M(2;0)(1)χ1M(2;0)(1)χ1M(2;0)(1)(...))))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)2)2}
1528FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(2,M(1,0)+1)) = ψ(ψM-I(2,M(1,0)+1)M-I(2,M(1,0)+1)(...)))
ψ(2-weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)+χM(1,1)(M(1,1)))
ψ(χ1M(2;0)(1)+χχ1M(2;0)(1)1M(2;0)(1)))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,1,,2(1,,2,,)2)1(1(1,,2,,)1,,1,,2(1,,2,,)2)2)2}
1529FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(3,M(1,0)+1)(0)) = ψ(M-I(2,M-I(2,...M-I(2,M(1,0)+1)...)))
ψ(1st fixed point of weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)2)
ψ(χ1M(2;0)(1)2)
{n,n(1(1,,2,,)1,,1,,1,,2(1,,2,,)2)2}
1530FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(3,M(1,0)+1)) = ψ(ψM-I(3,M(1,0)+1)M-I(3,M(1,0)+1)(...)))
ψ(2-weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)2M(1,1)(M(1,1)2))
ψ(χ1M(2;0)(1)2χ1M(2;0)(1)1M(2;0)(1)2))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1,,1,,1,,2(1,,2,,)2)1(1(1,,2,,)1,,1,,1,,2(1,,2,,)2)2)2}
1531FGH ψ(ψM(1,0))
FGH ψ(ψM(1,1))
FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(M(1,0),1)(0)) = ψ(ψM-I(M(1,0),M(1,0)+1)(0))
ψ(M(1,1)M(1,0))
ψ(χ1M(2;0)(1)χ1M(2;0)(0))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,1,,2)2(1,,2,,)2)2}
1532FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(M(1,0),1)) = ψ(M-I(M(1,0),M(1,0)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo-weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)M(1,0)M(1,1)(M(1,1)M(1,0)))
ψ(χ1M(2;0)(1)χ1M(2;0)(0)χ1M(2;0)(1)1M(2;0)(1)χ1M(2;0)(0)))
{n,n(1,,2(1,,2,,)1(1(1,,2,,)1(1(1,,2,,)1,,1,,2)2(1,,2,,)2)1(1(1,,2,,)1(1(1,,2,,)1,,1,,2)2(1,,2,,)2)2)2}
1533FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,0,M(1,0)+1)(0)) = ψ(M-I(M-I(M-I(...M-I(M(1,0),1),...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(M(1,1)M(1,1))
ψ(χ1M(2;0)(1)χ1M(2;0)(1))
{n,n(1(1,,2,,)1(1,,2,,)3)2}
1534FGH ψ(ψM(1,0))
ψ(cardinal)
FGH ψ(ψM(1,1))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,0,M(1,0)+1) = ψ(ψM-I(1,0,M(1,0)+1)M-I(1,0,M(1,0)+1)(...)))
ψ(M(1,1)M(1,1)M(1,1)(M(1,1)M(1,1)))
ψ(1st hyper-weakly inaccessible limit after 1st weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)(1)χ1M(2;0)(1)χ1M(2;0)(1)1M(2;0)(1)χ1M(2;0)(1)))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)3)2,,)1,,2)2}
1535FGH ψ(ψM(1,1))
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ΩM(1,1)+1) = ψ(χMM(1,1)+1(0)) = ψ(εM(1,0)+1) = ψ(M(1,0)M(1,0)M(1,0)M(1,0)...)
ψ(1st uncountable after 2nd weakly Mahlo limit of weakly Mahlo)
ψ(Ωχ1M(2;0)(1)+1) = ψ(χMχ1M(2;0)(1)+1(0)) = ψ(εχ1M(2;0)(0)+1) = ψ(χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)χ1M(2;0)(0)...)
{n,n(1,,2(1,,2,,)1(1,,2,,)3)2}
1536FGH ψ(ψM(1,1))
ψ(cardinal)
FGH ψ(ψM(1,2))
ψ(ψM(1,2)(cardinal))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,M(1,1)+1)(0)) = ψ(M-I(1,M-I(1,...M-I(1,M(1,1)+1)...)))
ψ(1st fixed point of weakly inaccessible limit after 2nd weakly Mahlo limit of weakly Mahlo)
ψ(M(1,2))
ψ(3d weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)(2))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)3)2}
1537FGH ψ(ψM(1,2))
ψ(cardinal)
FGH ψ(ψM(1,3))
ψ(ψM(1,3)(cardinal))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,M(1,2)+1)(0)) = ψ(M-I(1,M-I(1,...M-I(1,M(1,2)+1)...)))
ψ(1st fixed point of weakly inaccessible limit after 3d weakly Mahlo limit of weakly Mahlo)
ψ(M(1,3))
ψ(4-th weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)(3))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)4)2}
1538FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ω)) = ψ(M(1,sup(n)))
ψ(ω-th weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)(ω))
{n,n(1(1,,2,,)1(1,,2,,)1,2)2}
1539FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ω)×2)
ψ(χ1M(2;0)(ω)×2)
{n,n(1(1(1,,2,,)1(1,,2,,)1,2)2(1,,2,,)1(1,,2,,)1,2)2}
1540FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χMM(1,ω)+1(0)) = ψ(ΩM(1,ω)+1(0))
ψ(1st uncountable after ω-th weakly Mahlo limit of weakly Mahlo)
ψ(χMχ1M(2;0)(ω)+1(0)) = ψ(Ωχ1M(2;0)(ω)+1(0))
{n,n(1,,2(1,,2,,)1(1,,2,,)1,2)2}
1541FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(1,ω)+1)
ψ(1st weakly Mahlo after ω-th weakly Mahlo limit of weakly Mahlo)
ψ(Mχ1M(2;0)(ω)+1)
{n,n(1,,1,,2(1,,2,,)1(1,,2,,)1,2)2}
1542FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(1,ω)×2)
ψ(1st weakly Mahlo limit of weakly Mahlo-th weakly Mahlo after ω-th weakly Mahlo limit of weakly Mahlo)
ψ(Mχ1M(2;0)(ω)×2)
{n,n(1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,2)2(1,,2,,)1,2)2}
1543FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,ω+1)(0)(0)) = ψ(MMM...M(1,ω)+1)
ψ(1st fixed point of weakly Mahlo after ω-th weakly Mahlo limit of weakly Mahlo)
ψ(ψχχ1M(2;0)(ω+1)(ω+1)(ω+1)) = ψ(MMM...χ1M(2;0)(ω)+1)
{n,n(1(1,,2,,)1,,2(1,,2,,)1,2)2}
1544FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,ω)(0)) = ψ(ψχM(1,ω)(0)χM(1,ω)(0)χM(1,ω)(0)(...)))) = ψ(ψM-I(1,M(1,ω)+1)(0))
ψ(1st weakly inaccessible of weakly Mahlo after ω-th weakly Mahlo limit of weakly Mahlo)
ψ(χχ1M(2;0)(ω+1)(ω+1)) = ψ(ψχχ1M(2;0)(ω+1)(ω+1)χχ1M(2;0)(ω+1)(ω+1)χχ1M(2;0)(ω+1)(ω+1)(...))))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1,,2,,)1,2)1(1(1,,2,,)1,,2(1,,2,,)1,2)2(1,,2,,)2)2}
1545FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ω+1)) = ψ(χM(1,ω)M(1,ω)M(1,ω)(...)))) = ψ(ψM-I(2,M(1,ω)+1)(0))
ψ(χ1M(2;0)(ω+1)) = ψ(χχ1M(2;0)(ω+1)χ1M(2;0)(ω+1)χ1M(2;0)(ω+1)(...))))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)1,2)2}
1546FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ω+2)) = ψ(ψM-I(2,M(1,ω+1)+1)(0))
ψ(χ1M(2;0)(ω+2))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)2,2)2}
1547FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ω×2))
ψ(χ1M(2;0)(ω×2))
{n,n(1(1,,2,,)1(1,,2,,)1,3)2}
1548FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(1,ω)))
ψ(ω-th weakly Mahlo limit of weakly Mahlo-th weakly Mahlo limit of weakly Mahlo)
ψ(χ1M(2;0)1M(2;0)(ω)))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,2)2)2}
1549FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
ψ(ψM(2;0)(cardinal))
FGH ψ(ψK)
SAN
ψ(ψM(1,α)-I(1,0)(0)) = ψ(M(1,M(1,M(1,M(1,...)))))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo)
ψ(ψχM(2,0)(0)(0))
ψ(M(2;0)) = ψ(χ1M(2;0)1M(2;0)1M(2;0)(...))))
ψ(1st 2-weakly Mahlo)
ψ(χK(2,0)) = ψ(χK(2,2)) = ψ(χK(2,1)) =ψ(χK(2))
{n,n(1(1,,2,,)1(1,,2,,)1,,2)2}
1550FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(MψM(1,α)-I(1,0)(0)+1)
ψ(1st weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo)
ψ(MψχM(2,0)(0)(0)+1)
ψ(M(2;0)+M(1;ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1))
{n,n(1,,1,,2(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1551FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(MψM(1,α)-I(1,0)(0)×2)
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo-th weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo)
ψ(MψχM(2,0)(0)(0)×2)
ψ(M(2;0)+M(1;ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))×2))
{n,n(1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1552FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(1,ψM(1,α)-I(1,0)(0)+1)(0)) = ψ(MM...MψM(1,α)-I(1,0)(0)+1)
ψ(1st fixed point of (weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo))
ψ(ψχM(1,ψχM(2,0)(0)(0)+1)(1)(1)) = ψ(MM...MψχM(2,0)(0)(0)+1)
ψ(M(2;0)+ψχχ1M(2;0)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)) = ψ(M(2;0)+M(1;M(1;M(1;...M(1;ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)...)))))
{n,n(1(1,,2,,)1,,2(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1553FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,ψM(1,α)-I(1,0)(0)+1)) = ψ(ψM-I(1,ψM(1,α)-I(1,0)(0)+1)M-I(1,ψM(1,α)-I(1,0)(0)+1)(...)))
ψ(1st weakly inaccessible limit of (weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo))
ψ(χM(1,ψχM(2,0)(0)(0)+1)(1))
ψ(M(2;0)+χχ1M(2;0)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2)1(1(1,,2,,)1,,2)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1554FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(1,α)-I)

ψM(1,0)(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,ψM(1,α)-I(1,0)(0)+1)(0)) = ψ(M-I(1,M-I(1,...M-I(1,ψM(1,α)-I(1,0)(0)+1)...)))
ψ(1st fixed point of weakly inaccessible limit of (weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo))
ψ(M(1,ψM(1,α)-I(1,0)(0)+1))
ψ(1st weakly Mahlo limit of (weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo))
ψ(M(1,ψχM(2,0)(0)(0)+1))
ψ(M(2;0)+χ1M(2;0)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1555FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,ψM(1,α)-I(1,0)(0)×2))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo-th weakly Mahlo limit of (weakly Mahlo after 1st fixed point of weakly Mahlo limit of weakly Mahlo))
ψ(M(1,ψχM(2,0)(0)(0)×2))
ψ(M(2;0)+χ1M(2;0)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))×2))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1556FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(1,0)(1)) = ψ(M(1,M(1,M(1,...M(1,ψM(1,α)-I(1,0)(0)+1)...))))
ψ(2nd fixed point of weakly Mahlo limit of weakly Mahlo)
ψ(ψχM(2,0)(0)(1)) = ψ(M(1,M(1,M(1,...M(1,ψχM(2,0)(0)(0)+1)...))))
ψ(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0)+1)) = ψ(M(2;0)+χ1M(2;0)1M(2;0)1M(2;0)(...χ1M(2;0)χχ1M(2;0)(M(2;0))(M(2;0))(M(2;0))+1)))))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)3)2}
1557FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,0)) = ψ(ψM(1,α)-I(1,0)M(1,α)-I(1,0)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(χM(2,0)(0)) = ψ(ψχM(2,0)(0)χM(2,0)(0)(...)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0))) = ψ(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0))(M(2;0)+...)))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1558FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(MM(1,α)-I(1,0)+1)
ψ(1st weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(MχM(2,0)(0)+1)
ψ(M(2;0)+M(1;χχ1M(2;0)(M(2;0))(M(2;0))+1))
{n,n(1,,1,,2(1,,2,,)1(1,,2,,)1,,2)2}
1559FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(MM(1,α)-I(1,0)+ω)
ψ(ω-th weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(MχM(2,0)(0)+ω)
ψ(M(2;0)+M(1;χχ1M(2;0)(M(2;0))(M(2;0))+ω))
{n,n(1(1,,2,,)1,2(1,,2,,)1,,2)2}
1560FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(MMMM...M(1,α)-I(1,0)+1) = ψ(ψM-I(1,M(1,α)-I(1,0)+1)(0))
ψ(1st fixed point of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo))
ψ(ψχM(1,χM(2,0)(0)+1)(1)(1))
ψ(M(2;0)+ψχχ1M(2;0)χ1M(2;0)(M(2;0))(M(2;0))+1)χ1M(2;0)(M(2;0))(M(2;0))+1)χ1M(2;0)(M(2;0))(M(2;0))+1)) = ψ(M(2;0)+M(1;M(1;M(1;...M(1;χχ1M(2;0)(M(2;0))(M(2;0))+1)...)))))
{n,n(1(1,,2,,)1,,2(1,,2,,)1,,2)2}
1561FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M-I(1,M(1,α)-I(1,0)+1)) = ψ(ψM-I(1,M(1,α)-I(1,0)+1)M-I(1,M(1,α)-I(1,0)+1)(...)))
ψ(1st weakly inaccessible limit of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo))
ψ(χM(1,χM(2,0)(0)+1)(1))
ψ(M(2;0)+χχ1M(2;0)χ1M(2;0)(M(2;0))(M(2;0))+1)χ1M(2;0)(M(2;0))(M(2;0))+1))
{n,n(1(1,,2,,)1,,1(1(1,,2,,)1,,2(1,,2,,)1,,2)1(1(1,,2,,)1,,2(1,,2,,)1,,2)2(1,,2,,)1,,2)2}
1562FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(1,α)-I)

ψM(1,0)(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM-I(2,χM(2,0)(0)+1)(0)) = ψ(M-I(1,M-I(1,...M-I(1,M(1,α)-I(1,0)+1)...)))
ψ(M(1,M(1,α)-I(1,0)+1))
ψ(1st weakly Mahlo limit of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo))
ψ(M(1,χM(2,0)(0)+1))
ψ(M(2;0)+χ1M(2;0)χ1M(2;0)(M(2;0))(M(2;0))+1))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)1,,2)2}
1563FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(1,α)-I(1,0)+1))
ψ(ω-th weakly Mahlo limit of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo))
ψ(M(1,χM(2,0)(0)+ω))
ψ(M(2;0)+χ1M(2;0)χ1M(2;0)(M(2;0))(M(2;0))+ω))
{n,n(1(1,,2,,)1(1,,2,,)1,2,,2)2}
1564FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(1,1)(0)) = ψ(M(1,M(1,M(1,...M(1,α)-I(1,0)+1))))
ψ(1st fixed point of (weakly Mahlo limit of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)))
ψ(ψχM(2,0)(1)(1)) = ψ(M(1,M(1,M(1,...M(1,χM(2,0)(0)+1)...))))
ψ(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0)+1)(M(2;0)+1)) = ψ(M(2;0)+χ1M(2;0)1M(2;0)1M(2;0)(...χ1M(2;0)χ1M(2;0)(M(2;0))(M(2;0))+1)))))
{n,n(1(1,,2,,)1(1,,2,,)1,,3)2}
1565FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,1)) = ψ(ψM(1,α)-I(1,1)M(1,α)-I(1,1)(...)))
ψ(2nd weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(χM(2,0)(1)) = ψ(ψχM(2,0)(1)χM(2,0)(1)(...)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0)+1)) = ψ(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0)+1)(M(2;0)+ψχχ1M(2;0)(M(2;0))(M(2;0)+1)(M(2;0)+...)))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,3)1(1(1,,2,,)1(1,,2,,)1,,3)2)2}
1566FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,ω))
ψ(ω-th weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(χM(2,0)(ω))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0)+ω))
{n,n(1(1,,2,,)1(1,,2,,)1,,1,2)2}
1567FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,M(1,α)-I(1,0)))
ψ(1st weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo-th weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(χM(2,0)M(2,0)(0)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0))))
{n,n(1(1,,2,,)1(1,,2,,)1,,1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1568FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))
ψM(2,0)(cardinal))

FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(2,0)(0)) = ψ(M(1,α)-I(1,M(1,α)-I(1,M(1,α)-I(1,...))))
ψ(1st fixed point of weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0))
ψ(1st weakly Mahlo limit of weakly Mahlo limit of weakly Mahlo or 1st weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)+χ1M(2;0)(M(2;0))) = ψ(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0)+χχ1M(2;0)(M(2;0))(M(2;0)+...)))
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,2)2}
1569FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(2,0)) = ψ(ψM(1,α)-I(2,0)M(1,α)-I(2,0)(...)))
ψ(2-weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0)+χM(2,0)(M(2,0)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))1M(2;0)(M(2;0))))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,1,,2)1(1(1,,2,,)1(1,,2,,)1,,1,,2)2)2}
1570FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(3,0)(0)) = ψ(M(1,α)-I(2,M(1,α)-I(2,M(1,α)-I(2,...))))
ψ(1st fixed point of 2-weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0)2)
ψ(M(2;0)+χ1M(2;0)(M(2;0))2)
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,1,,2)2}
1571FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(3,0)) = ψ(ψM(1,α)-I(3,0)M(1,α)-I(3,0)(...)))
ψ(3-weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0)2M(2,0)(M(2,0)2))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))1M(2;0)(M(2;0))2))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,1,,1,,2)1(1(1,,2,,)1(1,,2,,)1,,1,,1,,2)2)2}
1572FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(1,0,0)(0)) = ψ(M(1,α)-I(M(1,α)-I(M(1,α)-I(...,0),0),0))
ψ(1st fixed point of α-weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0)M(2,0))
ψ(M(2;0)+χ1M(2;0)(M(2;0))χ1M(2;0)(M(2;0)))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)2}
1573FGH ψM(1,α)-I)
ψ(cardinal)
FGH ψM(2,0))

FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,0,0)) = ψ(ψM(1,α)-I(1,0,0)M(1,α)-I(1,0,0)(...)))
ψ(1st hyper-weakly inaccessible limit of weakly Mahlo limit of weakly Mahlo)
ψ(M(2,0)M(2,0)M(2,0)(M(2,0)M(2,0)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1574FGH ψM(1,α)-I)
FGH ψM(2,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(1,0,0,0,...)(0)) = ψ(ψM(1,α)-I(1ω)(0))
ψ(M(2,0)M(2,0)ω)
ψ(M(2;0)+χ1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))ω)
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)1,2,,)2)2}
1575FGH ψM(1,α)-I)
FGH ψM(2,0))
FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1,0,0,0,...)) = ψ(ψM(1,α)-I(1,0,0,0,...)M(1,α)-I(1,0,0,0,...)(...))) = ψ(M(1,α)-I(1ω))
ψ(M(2,0)M(2,0)ωM(2,0)(M(2,0)M(2,0)ω))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))ω))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)1,2,,)1,,2)2}
1576FGH ψM(1,α)-I)
FGH ψM(2,0))
FGH ψ(ψM(2;0))
SAN
ψ(ψM(1,α)-I(1M(1,α)-I(1M(1,α)-I(1M(1,α)-I(1...))))(0))
ψ(M(2,0)M(2,0)M(2,0))
ψ(M(2;0)+χ1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))χ1M(2;0)(M(2;0)))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)2,,)2)2}
1577FGH ψM(1,α)-I)
FGH ψM(2,0))
FGH ψ(ψM(2;0))
SAN
ψ(M(1,α)-I(1M(1,α)-I(1M(1,α)-I(1M(1,α)-I(1...)))))
ψ(M(2,0)M(2,0)M(2,0)M(2,0)(M(2,0)M(2,0)M(2,0)))
ψ(M(2;0)+χχ1M(2;0)(M(2;0))1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))χ1M(2;0)(M(2;0))))
{n,n(1(1,,2,,)1(1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)2,,)1,,2)2}
1578FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ΩM(2,0)+1) = ψ(χMM(2,0)+1(0)) = ψ(εM(2,0)+1) = ψ(M(2,0)M(2,0)M(2,0)M(2,0)...)
ψ(1st uncountable after 1st weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)+Ωχ1M(2;0)(M(2;0))+1) = ψ(χM(1;χ1M(2;0)(M(2;0))+1)(0))
{n,n(1,,2(1,,2,,)1(1,,2,,)1(1,,2,,)2)2}
1579FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(2,0)+1)
ψ(1st weakly Mahlo after 1st weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)+M(1;χ1M(2;0)(M(2;0))+1))
{n,n(1,,1,,2(1,,2,,)1(1,,2,,)1(1,,2,,)2)2}
1580FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(2,0)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)+χ1M(2;0)(M(2;0)+1))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)1(1,,2,,)2)2}
1581FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(2,1)(0)(0)) = ψ(M(1,M(1,...M(1,M(2,0)+1)...)))
ψ(1st fixed point of (weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo 2-limit of weakly Mahlo))
ψ(M(2;0)×2) = ψ(M(2;0)+M(2;0)) = ψ(M(2;0)+χ1M(2;0)(M(2;0)+χ1M(2;0)(M(2;0)+...)))
{n,n(1(1,,2,,)1(1,,2,,)1,,2(1,,2,,)2)2}
1582FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(2,1)(0)) = ψ(ψχM(2,1)(0)χM(2,1)(0)(...)))
ψ(1st weakly inaccessible limit of (weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo 2-limit of weakly Mahlo))
ψ(M(2;0)×2+χχ1M(2;0)(M(2;0))(M(2;0)×2))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)1(1(1,,2,,)1(1,,2,,)1,,2)2(1,,2,,)2)2}
1583FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(2,1)) = ψ(χM(2,1)M(2,1)(...)))
ψ(2nd weakly Mahlo limit of weakly Mahlo limit of weakly Mahlo or 2nd weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)×2+χ1M(2;0)(M(2;0)×2))
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,2(1,,2,,)2)2}
1584FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(2,ω))
ψ(ω-th weakly Mahlo limit of weakly Mahlo limit of weakly Mahlo or ω-th weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)×ω+χ1M(2;0)(M(2;0)×ω))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,2)2}
1585FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(3,0)(0)(0)) = ψ(M(1,M(1,M(1,M(1,...)))))
ψ(1st fixed point of weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)2) = ψ(M(2;0)×M(2;0)) = ψ(M(2;0)×χ1M(2;0)(M(2;0)×χ1M(2;0)(M(2;0)×...)))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,2)2}
1586FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(3,0)(0)) = ψ(ψχM(3,0)(0)χM(3,0)(0)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo 2-limit of weakly Mahlo)
ψ(M(2;0)2χ1M(2;0)(M(2;0))(M(2;0)2))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,2)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,2)2)2}
1587FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(3,0)) = ψ(χM(3,0)M(3,0)(...)))
ψ(1st weakly Mahlo limit of weakly Mahlo limit of weakly Mahlo limit of weakly Mahlo or 1st weakly Mahlo 3-limit of weakly Mahlo)
ψ(M(2;0)21M(2;0)(M(2;0)2))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,1,,2)2}
1588FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(4,0))
ψ(1st weakly Mahlo 4-limit of weakly Mahlo)
ψ(M(2;0)31M(2;0)(M(2;0)3))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,1,,2)2}
1589FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(5,0))
ψ(1st weakly Mahlo 5-limit of weakly Mahlo)
ψ(M(2;0)41M(2;0)(M(2;0)4))
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,1,,2)2}
1590FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,0)(0)(0)) = ψ(M(sup(n),0))
ψ(1st fixed point of weakly Mahlo n-limit of weakly Mahlo)
ψ(M(2;0)ω)
{n,n(1(2,,2,,)2)2}
1591FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,0)(0)(0)×2)
ψ(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω))
{n,n(1(1(2,,2,,)2)2(2,,2,,)2)2}
1592FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MψχM(ω,0)(0)(0)+1)
ψ(M(2;0)ω+M(1;ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1,,1,,2(2,,2,,)2)2}
1593FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ψχM(ω,0)(0)(0)+1))
ψ(M(2;0)ω1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1(1,,2,,)1,,1,,2(2,,2,,)2)2}
1594FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(2,ψχM(ω,0)(0)(0)+1))
ψ(M(2;0)ω1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+χ1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+1)))
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,2(2,,2,,)2)2}
1595FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,0)(0)(1)) = ψ(M(sup(n),ψχM(ω,0)(0)(0)+1))
ψ(2nd fixed point of weakly Mahlo n-limit of weakly Mahlo)
ψ(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω+1)) = ψ(M(2;0)ω1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+χ1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)+..)))
{n,n(1(2,,2,,)3)2}
1596FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,0)(0)χM(ω,0)(0)(0)))
ψ(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω)))
{n,n(1(2,,2,,)1(1(2,,2,,)2)2)2}
1597FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,0)(0))
ψ(1st weakly inaccessible limit of weakly Mahlo n-limit of weakly Mahlo)
ψ(M(2;0)ωχ1M(2;0)(M(2;0)ω)(M(2;0)ω))
{n,n(1(2,,2,,)1,,2)2}
1598FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MχM(ω,0)(0)+1)
ψ(M(2;0)ω+M(1;χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1,,1,,2(2,,2,,)1,,2)2}
1599FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,χM(ω,0)(0)+1))
ψ(M(2;0)ω1M(ω,0)χ1M(2;0)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1(1,,2,,)1,,1,,2(2,,2,,)1,,2)2}
1600FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,0)(1)(1)) = ψ(M(sup(n),χM(ω,0)(0)+1))
ψ(1st fixed point of (weakly Mahlo n-limit of (weakly Mahlo after 1st weakly inaccessible limit of weakly Mahlo n-limit of weakly Mahlo)))
ψ(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1)(M(2;0)ω+1)) = ψ(M(2;0)ω1M(2;0)χ1M(2;0)(M(2;0)ω)(M(2;0)ω)+χ1M(2;0)χ1M(2;0)(M(2;0)ω)(M(2;0)ω)+...)))
{n,n(1(2,,2,,)2,,2)2}
1601FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,0)(1))
ψ(2nd weakly inaccessible limit of weakly Mahlo n-limit of weakly Mahlo)
ψ(M(2;0)ωχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1))
{n,n(1(2,,2,,)1,,3)2}
1602FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,0)M(ω,0)(0)))
ψ(M(2;0)ωχ1M(2;0)(M(2;0)ω)(M(2;0)ωχ1M(2;0)(M(2;0)ω)(M(2;0)ω)))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,2)2)2}
1603FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)) = ψ(χM(ω,0)M(ω,0)M(ω,0)(...))))
ψ(1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2} = {n,n(1(2,,2,,)1,,1,,2)2}
1604FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(0)(0))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω))
{n,n(1(1(2,,2,,)2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1605FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(0)(1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω+1))
{n,n(1(1(2,,2,,)3)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1606FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(0)(M(ω,0)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))
{n,n(1(1(2,,2,,)1(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1607FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(0)) = ψ(M(ω,0)+ψχM(ω,0)(0)(M(ω,0)+ψχM(ω,0)(0)(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω)(M(2;0)ω+...)))
{n,n(1(1(2,,2,,)1,,2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1608FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+MχM(ω,0)(0)+1)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+M(1;χχ1M(2;0)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1(1(1,,2,,)2(2,,2,,)1,,2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1609FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+M(1,χM(ω,0)(0)+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χ1M(ω,0)χ1M(2;0)(M(2;0)ω)(M(2;0)ω)+1))
{n,n(1(1(1,,2,,)1(1,,2,,)2(2,,2,,)1,,2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1610FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(1)(1)) = ψ(M(ω,0)+M(sup(n),χM(ω,0)(0)+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1)(M(2;0)ω+1))
{n,n(1(1(2,,2,,)2,,2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1611FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(1)) = ψ(M(ω,0)+ψχM(ω,0)(1)(M(ω,0)+ψχM(ω,0)(1)(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1)) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1)(M(2;0)ωχχ1M(2;0)(M(2;0)ω)(M(2;0)ω+1)(M(2;0)ω+...)))
{n,n(1(1(2,,2,,)1,,3)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1612FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(2)(2)) = ψ(M(ω,0)+M(sup(n),χM(ω,0)(1)+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω+2)(M(2;0)ω+2))
{n,n(1(1(2,,2,,)2,,3)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1613FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(M(ω,0))(M(ω,0)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω))
{n,n(1(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1614FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+MψχM(ω,0)(M(ω,0))(M(ω,0))+1)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+M(1;ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1,,1,,2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1615FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+M(1,ψχM(ω,0)(M(ω,0))(M(ω,0))+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χ1M(2;0)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(1,,2,,)1,,1,,2(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2)2}
1616FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)))(M(ω,0)+1)) = ψ(M(ω,0)+M(sup(n),ψχM(ω,0)(M(ω,0))(M(ω,0))+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1)
{n,n(1(2,,2,,)2,,1(1(2,,2,,)1,,1,,2)2)2}
1617FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)))) = ψ(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)))(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)))(M(ω,0)+...))))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)))
{n,n(1(2,,2,,)1,,2(1(2,,2,,)1,,1,,2)2)2}
1618FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0))+1)(M(ω,0)+1)) = ψ(M(ω,0)+M(sup(n),χM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)))+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1)(M(2;0)ω1M(2;0)(M(2;0)ω)+1)
{n,n(1(2,,2,,)2,,2(1(2,,2,,)1,,1,,2)2)2}
1619FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0))+1)) = ψ(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0))+1)(M(ω,0)+ψχM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0)+1))(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(2,,2,,)1,,3(1(2,,2,,)1,,1,,2)2)2}
1620FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)χM(ω,0)(M(ω,0))(M(ω,0))×2))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)×2))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,2(1(2,,2,,)1,,1,,2)2)2(1(2,,2,,)1,,1,,2)2)2}
1621FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(M(ω,0))(M(ω,0)+1)) = ψ(M(ω,0)+χM(ω,0)(M(sup(n),ψχM(ω,0)(M(ω,0))(M(ω,0))+1)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)3)2}
1622FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(M(ω,0))) = ψ(M(ω,0)+ψχM(ω,0)(M(ω,0))(M(ω,0)+ψχM(ω,0)(M(ω,0))(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)1(1(2,,2,,)1,,1,,2)2)2}
1623FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+MχM(ω,0)(M(ω,0))+1)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+M(1;χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1,,1,,2(2,,2,,)1,,1,,2)2}
1624FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+M(1,χM(ω,0)(M(ω,0))+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χ1M(2;0)χ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(1,,2,,)1,,1,,2(2,,2,,)1,,1,,2)2}
1625FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(M(ω,0)+1)(M(ω,0)+1)) = ψ(M(ω,0)+M(sup(n),χM(ω,0)(M(ω,0))+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+1)(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(2,,2,,)2,,1,,2)2}
1626FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+ψχM(ω,0)(M(ω,0)+1)(M(ω,0)+2)) = ψ(M(ω,0)+M(sup(n),ψχM(ω,0)(M(ω,0)+1)(M(ω,0)+1)+1))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+ψχχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+1)(M(2;0)ω1M(2;0)(M(2;0)ω)+2))
{n,n(1(2,,2,,)3,,1,,2)2}
1627FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(M(ω,0)+1)) = ψ(M(ω,0)+ψχM(ω,0)(M(ω,0)+1)(M(ω,0)+ψχM(ω,0)(M(ω,0)+1)(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(2,,2,,)1,,2,,2)2}
1628FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(M(ω,0)+2)) = ψ(M(ω,0)+ψχM(ω,0)(M(ω,0)+2)(M(ω,0)+ψχM(ω,0)(M(ω,0)+2)(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+2))
{n,n(1(2,,2,,)1,,3,,2)2}
1629FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)+χM(ω,0)(M(ω,0)+χM(ω,0)(M(ω,0))))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω))))
{n,n(1(2,,2,,)1,,1(1(2,,2,,)1,,1,,2)2,,2)2}
1630FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)×2) = ψ(M(ω,0)+M(ω,0)) = ψ(M(ω,0)+χM(ω,0)(M(ω,0)+χM(ω,0)(M(ω,0)+...)))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)×2) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χ1M(2;0)(M(2;0)ω)) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+χχ1M(2;0)(M(2;0)ω)(M(2;0)ω1M(2;0)(M(2;0)ω)+...)))
{n,n(1(2,,2,,)1,,1,,3)2}
1631FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)2)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)2)
{n,n(1(2,,2,,)1,,1,,1,,2)2}
1632FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0)M(ω,0))
ψ(M(2;0)ω1M(2;0)(M(2;0)ω)χ1M(2;0)(M(2;0)ω))
{n,n(1(2,,2,,)1(1,,2,,)2)2}
1633FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ΩM(ω,0)+1) = ψ(χMM(ω,0)+1(0)) = ψ(εM(ω,0)+1) = ψ(M(ω,0)M(ω,0)M(ω,0)M(ω,0)...)
ψ(1st uncountable after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ωM(1;M(2;0)ω1M(2;0)(M(2;0)ω)+1)(0)) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω)χ1M(2;0)(M(2;0)ω)...)
{n,n(1(2,,2,,)1(1,,2(1,,2,,)2)2)2}
1634FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(MM(ω,0)+1)
ψ(1st weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+M(1;M(2;0)ω1M(2;0)(M(2;0)ω)+1))
{n,n(1(2,,2,,)1(1,,1,,2(1,,2,,)2)2)2}
1635FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,(M(ω,0)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω1M(2;0)(M(2;0)ω+1))
{n,n(1(2,,2,,)1(1(1,,2,,)1,,2)2)2}
1636FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(2;M(ω,0)(0))(0)) = ψ(M(1,M(1,M(1,(...M(ω,0)+1...))))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+M(2;0)) = ψ(M(2;0)ω1M(2;0)(M(2;0)ω1M(2;0)(M(2;0)ω1M(2;0)(M(2;0)ω+...))))
{n,n(1(2,,2,,)1(1(1,,2,,)1(1,,2,,)2)2)2}
1637FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(2,(M(ω,0)+1))
ψ(M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)ω+M(2;0)))
ψ(1st weakly Mahlo 2-limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
{n,n(1(2,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2)2)2}
1638FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(3;M(ω,0)(0))(0)) = ψ(M(2,M(2,M(2,(...M(ω,0)+1...))))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+M(2;0)2) = ψ(M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)ω+M(2;0)+...))))
{n,n(1(2,,2,,)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)2)2)2}
1639FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(3,(M(ω,0)+1))
ψ(1st weakly Mahlo 2-limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+M(2;0)21M(2;0)(M(2;0)ω+M(2;0)2))
{n,n(1(2,,2,,)1(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,2)2)2}
1640FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,1)(0)(0)) = ψ(M(sup(n),M(ω,0)+1))
ψ(1st fixed point of weakly Mahlo n-limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω×2) = ψ(M(2;0)ω+M(2;0)ω)
{n,n(1(2,,2,,)1(1(2,,2,,)2)2)2}
1641FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,1)(0)) = ψ(ψχM(ω,1)(0)χM(ω,1)(0)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo n-limit of weakly Mahlo after 1st weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω×2+χχ1M(2;0)(M(2;0)ω×2)(M(2;0)ω×2))
{n,n(1(2,,2,,)1,,2(1,,2,,)2)2}
1642FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,1)) = ψ(χM(ω,1)M(ω,1)(...)))
ψ(2nd weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω×2+χ1M(2;0)(M(2;0)ω×2))
{n,n(1(2,,2,,)1,,1,,2(1,,2,,)2)2}
1643FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,2))
ψ(3d weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω×3+χ1M(2;0)(M(2;0)ω×3))
{n,n(1(2,,2,,)1,,1,,2(1,,2,,)3)2}
1644FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω+1,0)(0)(0)) = ψ(M(ω,M(ω,M(ω,...))))
ψ(1st fixed point of weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+1)
{n,n(1(2,,2,,)1(1(2,,2,,)2)2(1,,2,,)2)2}
1645FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω+1,0)(0)) = ψ(ψχM(ω+1,0)(0)χM(ω+1,0)(0)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+1χ1M(2;0)(M(2;0)ω+1)(M(2;0)ω+1))
{n,n(1(2,,2,,)1(1,,2,,)1,,2)2}
1646FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(ω+1,0)) = ψ(χM(ω+1,0)M(ω+1,0)(...)))
ψ(1st weakly Mahlo limit of weakly Mahlo ω-limit of weakly Mahlo)
ψ(M(2;0)ω+11M(2;0)(M(2;0)ω+1))
{n,n(1(2,,2,,)1(1,,2,,)1,,1,,2)2}
1647FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω+2,0))
ψ(M(2;0)ω+21M(2;0)(M(2;0)ω+2))
{n,n(1(2,,2,,)1(1,,2,,)1(1,,2,,)1,,1,,2)2}
1648FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω×2,0))
ψ(M(2;0)ω×21M(2;0)(M(2;0)ω×2))
{n,n(1(2,,2,,)1(2,,2,,)1,,1,,2)2}
1649FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω2,0))
ψ(M(2;0)ω21M(2;0)(M(2;0)ω2))
{n,n(1(3,,2,,)1,,1,,2)2}
1650FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(Ω,0))
ψ(1st weakly Mahlo 1st uncontable-limit of weakly Mahlo)
ψ(M(2;0)Ω1M(2;0)(M(2;0)Ω))
{n,n(1(1'2,,2,,)1,,1,,2)2}
1651FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(M,0))
ψ(1st weakly Mahlo 1st weakly Mahlo-limit of weakly Mahlo)
ψ(M(2;0)M1M(2;0)(M(2;0)M))
{n,n(1(1(1(1,,2,,)2)2,,2,,)1,,1,,2)2}
1652FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(M(M,0),0))
ψ(1st weakly Mahlo 1st weakly Mahlo-limit of weakly Mahlo-limit of weakly Mahlo)
ψ(M(2;0)χ1M(2;0)(M(2;0)M)1M(2;0)(M(2;0)χ1M(2;0)(M(2;0)M))
{n,n(1(1(1(1(1(1,,2,,)2)2,,2,,)1(1,,2,,)2)2,,2,,)1,,1,,2)2}
1653FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0)(0)(0)) = ψ(M(M(M(...,0),0),0))
ψ(1st weakly Mahlo (1st fixed point of α)-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)) = ψ(M(2;0)χ1M(2;0)(M(2;0)χ1M(2;0)(M(2;0)...)))
{n,n(1(1(1(1,,3,,)2)2,,2,,)2)2} = {n,n(1(1,,3,,)2)2}
1654FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MψχM(1,0,0)(0)(0)+1)
ψ(M(2;0)M(2;0)+M(1;ψχχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1))
{n,n(1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1655FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,ψχM(1,0,0)(0)(0)+1))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1))
{n,n(1(1,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1656FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ψχM(1,0,0)(0)(0),1)(1)(1)) = ψ(M(1,M(1,...M(1,ψχM(1,0,0)(0)(0)+1)...)))
ψ(M(2;0)M(2;0)χχ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1657FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MψχM(ψχM(1,0,0)(0)(0),1)(1)(1)+1)
ψ(M(2;0)M(2;0)+M(1;ψχχ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)+1))
{n,n(1,,1,,2(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1658FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ψχM(1,0,0)(0)(0),1)(1)(2)) = ψ(M(1,M(1,...M(1,ψχM(ψχM(1,0,0)(0)(0),1)(1)(1)+1)...)))
ψ(M(2;0)M(2;0)χχ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)3(1(1(1,,3,,)2)2,,2,,)2)2}
1659FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ψχM(1,0,0)(0)(0),1)(1)) = ψ(ψχM(ψχM(1,0,0)(0)(0),1)(1)χM(ψχM(1,0,0)(0)(0),1)(1)(...)))
ψ(M(2;0)M(2;0)χ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1660FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ψχM(1,0,0)(0)(0),1)) = ψ(χM(ψχM(1,0,0)(0)(0),1)M(ψχM(1,0,0)(0)(0),1)(...)))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+1))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1661FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ψχM(1,0,0)(0)(0),2)(2)(2)) = ψ(M(1,M(1,...M(ψχM(1,0,0)(0)(0),1)+1...)))
ψ(M(2;0)M(2;0)χχ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1662FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ψχM(1,0,0)(0)(0),2)(2)) = ψ(ψχM(ψχM(1,0,0)(0)(0),2)(2)χM(ψχM(1,0,0)(0)(0),2)(2)(...)))
ψ(M(2;0)M(2;0)χ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,2(1,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1663FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ψχM(1,0,0)(0)(0),2)) = ψ(χM(ψχM(1,0,0)(0)(0),2)M(ψχM(1,0,0)(0)(0),2)(...)))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+2))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,1,,2(1,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1664FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ψχM(1,0,0)(0)(0)+1,0)(0)(0)) = ψ(M(ψχM(1,0,0)(0)(0),M(ψχM(1,0,0)(0)(0),...)))
ψ(M(2;0)M(2;0)χχ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0))χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0))χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0)))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,1(1(1(1,,3,,)2)2,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2}
1665FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ψχM(1,0,0)(0)(0)+1,0)(0)) = ψ(ψχM(ψχM(1,0,0)(0)(0)+1,0)(0)χM(ψχM(1,0,0)(0)(0)+1,0)(0)(...)))
ψ(M(2;0)M(2;0)χ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0))χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0)))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,1(1,,2,,)1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1666FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ψχM(1,0,0)(0)(0)+1,0)) = ψ(χM(ψχM(1,0,0)(0)(0)+1,0)M(ψχM(1,0,0)(0)(0)+1,0)(...)))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0)))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1(1,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1667FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ψχM(1,0,0)(0)(0)2,0))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))2))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)2)3,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1668FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(M(ψχM(1,0,0)(0)(0)+1,0),0))
ψ(M(2;0)M(2;0)1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))χ1M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0))+M(2;0))))
{n,n(1(1(1(1(1(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)2,,2,,)2)2,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1669FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0)(0)(1)) = ψ(M(M(...M(ψχM(1,0,0)(0)(0)+1,0)...,0),0))
ψ(1st weakly Mahlo (2nd fixed point of α)-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)3)2}
1670FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0)(0)) = ψ(ψχM(1,0,0)(0)χM(1,0,0)(0)(...)))
ψ(1st weakly Mahlo (1st weakly inaccessible limit of α)-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)χ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0)))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,2)2}
1671FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0)(1)(1)) = ψ(M(M(...M(χM(1,0,0)(0)+1,0)...,0),0))
ψ(1st weakly Mahlo (1st weakly Mahlo (1st fixed point of α)-limit of weakly Mahlo after 1st weakly inaccessible limit of α)-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)χχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0)+1)(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)2,,2)2}
1672FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0)(1)) = ψ(ψχM(1,0,0)(1)χM(1,0,0)(1)(...)))
ψ(1st weakly Mahlo (2nd weakly inaccessible limit of α)-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)χ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,3)2}
1673FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0)) = ψ(χM(1,0,0)M(1,0,0)(...)))
ψ(1st weakly Mahlo hyper-limit of weakly Mahlo or 1st weakly Mahlo 1,0-hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)1M(2;0)(M(2;0)M(2;0)))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,1,,2)2}
1674FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,1)(0)(0)) = ψ(M(M(M(...M(1,0,0)+1...,0),0),0))
ψ(1st weakly Mahlo (1st fixed point of α)-limit of weakly Mahlo after 1st weakly Mahlo hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)×2))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1(1,,2,,)2)2}
1675FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,1)(0)) = ψ(ψχM(1,0,1)(0)χM(1,0,1)(0)(...)))
ψ(1st weakly Mahlo (1st weakly inaccessible limit of α)-limit of weakly Mahlo after 1st weakly Mahlo hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)×2+χχ1M(2;0)(M(2;0)M(2;0)×2)(M(2;0)M(2;0)×2))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,2(1,,2,,)2)2}
1676FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,1)) = ψ(χM(1,0,1)M(1,0,1)(...)))
ψ(2nd weakly Mahlo hyper-limit of weakly Mahlo or 2nd weakly Mahlo 1,0-hyper-weakly inaccessible)
ψ(M(2;0)M(2;0)×2+χ1M(2;0)(M(2;0)M(2;0)×2))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,1,,2(1,,2,,)2)2}
1677FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,1,0)(0)(0)) = ψ(M(1,0,M(1,0,M(1,0,...))))
ψ(1st fixed point of weakly Mahlo hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,1,,2(1(1(1,,3,,)2)2,,2,,)2)2}
1678FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,1,0)(0)) = ψ(ψχM(1,1,0)(0)χM(1,1,0)(0)(...)))
ψ(1st weakly inaccessible limit of weakly Mahlo hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)+1χ1M(2;0)(M(2;0)M(2;0)+1)(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1,,1(1,,2,,)2,,2)2}
1679FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(1,1,0)) = ψ(χM(1,1,0)M(1,1,0)(...)))
ψ(1st weakly Mahlo 2-hyper-limit of weakly Mahlo or 1st weakly Mahlo 1,1-hyper-weakly inaccessible)
ψ(M(2;0)M(2;0)+11M(2;0)(M(2;0)M(2;0)+1))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1(1,,2,,)1,,1,,2)2}
1680FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(2,0,0)(0)(0)) = ψ(M(1,M(1,M(1,...,0),0),0))
ψ(1st weakly Mahlo (1st fixed point of α)-hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)×2)
{n,n(1(1(1(1,,3,,)2)2,,2,,)1(1(1(1,,3,,)2)2,,2,,)2)2}
1681FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(χM(2,0,0)(0)) = ψ(ψχM(2,0,0)(0)χM(2,0,0)(0)(...)))
ψ(1st weakly Mahlo (1st weakly inaccessible limit of α)-hyper-limit of weakly Mahlo)
ψ(M(2;0)M(2;0)×2χ1M(2;0)(M(2;0)M(2;0)+1)(M(2;0)M(2;0)×2))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1(1(1(1,,3,,)2)2,,2,,)1,,2)2}
1682FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(2,0,0)) = ψ(ψχM(2,0,0)(0)χM(2,0,0)(0)(...)))
ψ(1st weakly Mahlo hyper-hyper-limit of weakly Mahlo or 1st weakly Mahlo 2,0-hyper-weakly inaccessible)
ψ(M(2;0)M(2;0)×21M(2;0)(M(2;0)M(2;0)×2))
{n,n(1(1(1(1,,3,,)2)2,,2,,)1(1(1(1,,3,,)2)2,,2,,)1,,1,,2)2}
1683FGH ψ-function
ψ(cardinal)
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,0,0))
ψ(1st weakly Mahlo hyperω-limit of weakly Mahlo or 1st weakly Mahlo ω,0-hyper-weakly inaccessible)
ψ(M(2;0)M(2;0)×ω1M(2;0)(M(2;0)M(2;0)×ω))
{n,n(1(2(1(1,,3,,)2)2,,2,,)1,,1,,2)2}
1684FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(M(1,0,0),0,0))
ψ(M(2;0)M(2;0)×χ1M(2;0)(M(2;0)M(2;0))1M(2;0)(M(2;0)M(2;0)×χ1M(2;0)(M(2;0)M(2;0))))
{n,n(1(1(1(1(1(1,,3,,)2)2,,2,,)1(1,,2,,)2)2(1(1,,3,,)2)2,,2,,)1,,1,,2)2}
1685FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0)(0)(0)) = ψ(M(M(M(...,0,0),0,0),0,0))
ψ(M(2;0)M(2;0)2) = ψ(M(2;0)M(2;0)×M(2;0)) = ψ(M(2;0)M(2;0)×χ1M(2;0)(M(2;0)M(2;0)×χ1M(2;0)(M(2;0)M(2;0)×...)))
{n,n(1(1(1(1,,3,,)2)3,,2,,)2)2}
1686FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0)(0)(1)) = ψ(M(M(M(...ψχM(1,0,0,0)(0)(0)+1...,0,0),0,0),0,0))
ψ(M(2;0)M(2;0)2χχ1M(2;0)(M(2;0)M(2;0)2)(M(2;0)M(2;0)2)(M(2;0)M(2;0)2+1))
{n,n(1(1(1(1,,3,,)2)3,,2,,)3)2}
1687FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0,0)(0)) = ψ(ψχM(1,0,0,0)(0)χM(1,0,0,0)(0)(...)))
ψ(M(2;0)M(2;0)2χ1M(2;0)(M(2;0)M(2;0)2)(M(2;0)M(2;0)2))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1,,2)2}
1688FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0)(1)(1)) = ψ(M(M(...M(χM(1,0,0,0)(0)+1,0)...,0),0))
ψ(M(2;0)M(2;0)2χχ1M(2;0)(M(2;0)M(2;0)2)(M(2;0)M(2;0)2+1)(M(2;0)M(2;0)2+1))
{n,n(1(1(1(1,,3,,)2)3,,2,,)2,,2)2}
1689FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0,0)(1)) = ψ(M(M(M(...ψχM(1,0,0,0)(1)(1)+1...,0,0),0,0),0,0))
ψ(M(2;0)M(2;0)2χ1M(2;0)(M(2;0)M(2;0)2)(M(2;0)M(2;0)2+1))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1,,3)2}
1690FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0,0)) = ψ(χM(1,0,0,0)M(1,0,0,0)(...)))
ψ(M(2;0)M(2;0)21M(2;0)(M(2;0)M(2;0)2))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1,,1,,2)2}
1691FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0,1))
ψ(M(2;0)M(2;0)2×2+χ1M(2;0)(M(2;0)M(2;0)2×2))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1,,1,,2(1,,2,,)2)2}
1692FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,1,0))
ψ(M(2;0)M(2;0)2+11M(2;0)(M(2;0)M(2;0)2+1))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1(1,,2,,)1,,1,,2)2}
1693FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,1,0,0))
ψ(M(2;0)M(2;0)2+M(2;0)1M(2;0)(M(2;0)M(2;0)2+M(2;0)))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1(1(1(1,,3,,)2)2,,2,,)1,,1,,2)2}
1694FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(2,0,0,0))
ψ(M(2;0)M(2;0)2×21M(2;0)(M(2;0)M(2;0)2×2))
{n,n(1(1(1(1,,3,,)2)3,,2,,)1(1(1(1,,3,,)2)3,,2,,)1,,1,,2)2}
1695FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0,0,0))
ψ(M(2;0)M(2;0)31M(2;0)(M(2;0)M(2;0)3))
{n,n(1(1(1(1,,3,,)2)4,,2,,)1,,1,,2)2}
1696FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0,0,0,0))
ψ(M(2;0)M(2;0)41M(2;0)(M(2;0)M(2;0)4))
{n,n(1(1(1(1,,3,,)2)5,,2,,)1,,1,,2)2}
1697FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0,0,...)(0)(0)) = ψ(ψχM(1ω)(0)(0))
ψ(M(2;0)M(2;0)ω)
{n,n(1(1(1(1,,3,,)2)1,2,,2,,)2)2}
1698FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0,0,...)(0)(0)+ψχM(1,0,0)(0)(0))
ψ(M(2;0)M(2;0)ωχχ1M(2;0)(M(2;0)M(2;0))(M(2;0)M(2;0))(M(2;0)M(2;0)))
{n,n(1(1(1(1(1,,3,,)2)2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1699FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0,0,...)(0)(0)×2)
ψ(M(2;0)M(2;0)ωχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω))
{n,n(1(1(1(1(1,,3,,)2)1,2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1700FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ΩψχM(1,0,0,0,0,...)(0)(0)+1)
ψ(M(2;0)M(2;0)ωψχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)+1)
{n,n(1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1701FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χMψχM(1,0,0,0,0,...)(0)(0)+1(M(2;0)M(2;0)ω) = ψ(IψχM(1,0,0,0,0,...)(0)(0)+1)
ψ(M(2;0)M(2;0)ωM(1;ψχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)+1)(M(2;0)M(2;0)ω))
{n,n(1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1702FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MψχM(1,0,0,0,0,...)(0)(0)+1)
ψ(M(2;0)M(2;0)ω+M(1;ψχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)+1))
{n,n(1,,1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1703FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(MψχM(1,0,0,0,0,...)(0)(0)×2)
ψ(M(2;0)M(2;0)ω+M(1;ψχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)×2))
{n,n(1(1,,2,,)1(1(1(1(1,,3,,)2)1,2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1704FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0,0,...)(0)(1)) = ψ(ψχM(1,0,0,0,0,...ψχM(1,0,0,0,0,...)(0)(0)+1)(0)(0))
ψ(M(2;0)M(2;0)ωχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω+1))
{n,n(1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1705FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0,0,0,...)(0)) = ψ(χM(1ω)(0))
ψ(M(2;0)M(2;0)ωχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1706FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,0,0,...)(1)(2))
ψ(M(2;0)M(2;0)ωχχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω+1)(M(2;0)M(2;0)ω+2))
{n,n(1(1,,2,,)1,,3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1707FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,0,0,0,...)(1))
ψ(M(2;0)M(2;0)ωχ1M(2;0)(M(2;0)M(2;0)ω)(M(2;0)M(2;0)ω+1))
{n,n(1(1,,2,,)1(1(1,,2,,)1,,3(1(1(1,,3,,)2)1,2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1708FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,0,0,0,...)) = ψ(M(1ω))
ψ(M(2;0)M(2;0)ω1M(2;0)(M(2;0)M(2;0)ω))
{n,n(1(1,,2,,)1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1709FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(1,0,0,0,0,...)+1))
ψ(M(2;0)M(2;0)ω1M(2;0)(M(2;0)M(2;0)ω+1))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1710FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(2,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0))
{n,n(1(1,,2,,)1(1,,2,,)1(1(1,,2,,)1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1711FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(2,M(1,0,0,0,0,...)+1)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)+χχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0))(M(2;0)M(2;0)ω+M(2;0))))
{n,n(1(1,,2,,)1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1712FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(2,M(1,0,0,0,0,...)+1)(1))
ψ(M(2;0)M(2;0)ω+M(2;0)+χχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0))(M(2;0)M(2;0)ω+M(2;0)+1)))
{n,n(1(1,,2,,)1(1,,2,,)1,,3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1713FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(2,M(1,0,0,0,0,...)+1))
ψ(M(2;0)M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)))
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1714FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(2,M(1,0,0,0,0,...)+1)))
ψ(M(2;0)M(2;0)ω+M(2;0)+χ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)+1))
{n,n(1(1,,2,,)1,,1,,2(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1715FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(2,M(1,0,0,0,0,...)+2)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)×2)
{n,n(1(1,,2,,)1(1,,2,,)1,,1,,2(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1716FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(3,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)2)
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1717FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)ω)
{n,n(1(2,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1718FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω,M(1,0,0,0,0,...)+1)(0)(1))
ψ(M(2;0)M(2;0)ω+M(2;0)ωχχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)ω)(M(2;0)M(2;0)ω+M(2;0)ω)(M(2;0)M(2;0)ω+M(2;0)ω+1))
{n,n(1(2,,2,,)3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1719FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,M(1,0,0,0,0,...)+1)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)ωχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)ω)(M(2;0)M(2;0)ω+M(2;0)ω))
{n,n(1(2,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1720FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(ω,M(1,0,0,0,0,...)+1)(1))
ψ(M(2;0)M(2;0)ω+M(2;0)ωχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)ω)(M(2;0)M(2;0)ω+M(2;0)ω+1))
{n,n(1(2,,2,,)1,,3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1721FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω,M(1,0,0,0,0,...)+1))
ψ(M(2;0)M(2;0)ω+M(2;0)ω+M(2;0)ω1M(2;0)(M(2;0)M(2;0)ω+M(2;0)ω))
{n,n(1(2,,2,,)1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1722FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(ω+1,M(1,0,0,0,0,...)+1))
ψ(M(2;0)M(2;0)ω+M(2;0)ω+11M(2;0)(M(2;0)M(2;0)ω+M(2;0)ω+1))
{n,n(1(2,,2,,)1(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1723FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(ω2,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)ω2)
{n,n(1(3,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1724FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0))
{n,n(1(1(1,,3,,)2)2,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1725FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,M(1,0,0,0,0,...)+1)(0)(1))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)χχ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)M(2;0))(M(2;0)M(2;0)ω+M(2;0)M(2;0))(M(2;0)M(2;0)ω+M(2;0)M(2;0)+1))
{n,n(1(1(1,,3,,)2)2,,2,,)3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1726FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,M(1,0,0,0,0,...)+1)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)χ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)M(2;0))(M(2;0)M(2;0)ω+M(2;0)M(2;0)))
{n,n(1(1(1,,3,,)2)2,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1727FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1,0,M(1,0,0,0,0,...)+1)(1))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)χ1M(2;0)(M(2;0)M(2;0)ω+M(2;0)M(2;0))(M(2;0)M(2;0)ω+M(2;0)M(2;0)+1))
{n,n(1(1(1,,3,,)2)2,,2,,)1,,3(1(1(1,,3,,)2)1,2,,2,,)2)2}
1728FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,0,M(1,0,0,0,0,...)+1))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)1M(2;0)(M(2;0)M(2;0)ω+M(2;0)M(2;0)))
{n,n(1(1(1,,3,,)2)2,,2,,)1,,1,,2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1729FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1,M(1,0,M(1,0,0,0,0,...)+1)))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)1M(2;0)(M(2;0)M(2;0)ω+M(2;0)M(2;0)+1))
{n,n(1(1(1,,3,,)2)2,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1730FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1,0,0,M(1,0,0,0,0,...)+1)(0)(0))
ψ(M(2;0)M(2;0)ω+M(2;0)M(2;0)2)
{n,n(1(1(1,,3,,)2)3,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)2)2}
1731FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1ω10)(0)(0))
ψ(M(2;0)M(2;0)ω×2)
{n,n(1(1(1(1,,3,,)2)1,2,,2,,)3)2}
1732FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1ω10)(0))
ψ(M(2;0)M(2;0)ω×2+χχ1M(2;0)(M(2;0)M(2;0)ω×2)(M(2;0)M(2;0)ω×2))
{n,n(1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)3)2}
1733FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1ω10))
ψ(M(2;0)M(2;0)ω×2+χ1M(2;0)(M(2;0)M(2;0)ω×2))
{n,n(1(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)3)2}
1734FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1ω11)(0)(0))
ψ(M(2;0)M(2;0)ω+1)
{n,n(1(1(1(1,,3,,)2)1,2,,2,,)1,,2)2}
1735FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1ω11)(0))
ψ(M(2;0)M(2;0)ω+1χ1M(2;0)(M(2;0)M(2;0)ω+1)(M(2;0)M(2;0)ω+1))
{n,n(1(1,,2,,)1,,2(1(1(1,,3,,)2)1,2,,2,,)1,,2)2}
1736FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1ω11))
ψ(M(2;0)M(2;0)ω+11M(2;0)(M(2;0)M(2;0)ω+1))
{n,n(1(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1,2,,2,,)1,,2)2}
1737FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1ω+1)(0)(0))
ψ(M(2;0)M(2;0)ω+1)
{n,n(1(1(1(1,,3,,)2)2,2,,2,,)2)2}
1738FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1Ω)(0)(0))
ψ(M(2;0)M(2;0)Ω)
{n,n(1(1(1(1,,3,,)2)1'2,,2,,)2)2}
1739FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1M)(0)(0))
ψ(M(2;0)M(2;0)M)
{n,n(1(1(1(1,,3,,)2)1(1,,1,,2)2,,2,,)2)2}
1740FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(ψχM(1M(1M(1M(1...))))(0)(0))
ψ(M(2;0)M(2;0)M(2;0))
{n,n(1(1(1(1,,3,,)2)1(1(1,,3,,)2)2,,2,,)2)2}
1741FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(χM(1M(1M(1M(1...))))(0))
ψ(M(2;0)M(2;0)M(2;0)χ1M(2;0)(M(2;0)M(2;0)M(2;0)+1)(M(2;0)M(2;0)M(2;0)))
{n,n(1(1,,2,,)1,,2(1(1(1,,3,,)2)1(1(1,,3,,)2)2,,2,,)2)2}
1742FGH ψ-function
FGH ψ(ψM(2;0))
SAN
ψ(M(1M(1M(1M(1...)))))
ψ(M(2;0)M(2;0)M(2;0)1M(2;0)(M(2;0)M(2;0)M(2;0)))
{n,n(1(1,,2,,)1(1,,2,,)2(1(1(1,,3,,)2)1(1(1,,3,,)2)2,,2,,)2)2}
1743FGH ψ-function
SAN
ψ(M(2;0)M(2;0)M(2;0)M(2;0))
{n,n(1(1(1,,3,,)2)2(1,,3,,)2)2}
1744FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;0)M(2;0)M(2;0)M(2;0)...) = ψ(εM(2;0)+1) = ψ(ΩM(2;0)+1)
ψ(1st uncountable after 1st 2-weakly Mahlo )
{n,n(1,,2(1,,3,,)2)2}
1745FGH ψ-function
ψ(cardinal)
SAN
ψ(M(1;M(2;0)+1)) = (ψ(MM(2;0)+1)
ψ(1st weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)2(1,,3,,)2)2}
1746FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1M(2;1)(0)(0)(0)) = ψ(M(1;M(1;M(1;M(1;...M(2;0)+1...))))))
ψ(1st fixed point of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1,,3,,)2)2(1,,3,,)2)2}
1747FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1M(2;1)(0)(0)) = ψ(M-I(1,M(2;0)+1))
ψ(1st weakly inaccessible limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1,,2(1,,3,,)2)2}
1748FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1M(2;1)(0)(1)) = ψ(M-I(1,M(2;0)+2))
ψ(2nd weakly inaccessible limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1,,3(1,,3,,)2)2}
1749FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1M(2;1)(0)) = ψ(M(1,M(2;0)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1,,2,,)2(1,,3,,)2)2}
1750FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1M(2;1)(1)) = ψ(M(1,M(2;0)+2))
ψ(2nd weakly Mahlo limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1,,2,,)3(1,,3,,)2)2}
1751FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(2;1)) = ψ(M(1,M(1,M(1,M(1,...M(2;0)+1...)))))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo after 1st 2-weakly Mahlo)
ψ(2nd 2-weakly Mahlo)
ψ(χχK(3)(2,3)) = ψ(χK(2,3)
{n,n(1(1,,2,,)1(1,,2,,)1,,2(1,,3,,)2)2}
1752FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;1)+χ1M(2;1)(M(2;1))) = ψ(M(2,M(2;0)+1)))
ψ(1st weakly Mahlo 2-limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)2(1,,3,,)2)2}
1753FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;1)×2) = ψ(M(1,M(1,...M(1,M(2,M(2;0)+1)+1)...)))
ψ(1st fixed point of (weakly Mahlo limit of weakly Mahlo after 1st weakly Mahlo 2-limit of weakly Mahlo) after 1st 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)3(1,,3,,)2)2}
1754FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;1)M(2;1)) = ψ(M(M(M(...M(1,M(2;0)+1)...,0),0),0))
ψ(1st weakly Mahlo (1st fixed point of α)-limit of weakly Mahlo after 1st 2-weakly Mahlo)
{n,n(1(1(1(1,,3,,)3)2,,2,,)2(1,,3,,)2)2}
1755FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;1)M(2;1)M(2;1)M(2;1)...) = ψ(εM(2;1)+1) = ψ(ΩM(2;1)+1)
ψ(1st uncountable after 2nd 2-weakly Mahlo )
{n,n(1,,2(1,,3,,)3)2}
1756FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1M(2;2)(0)(0)(0)) = ψ(M(1;M(1;M(1;M(1;...M(2;1)+1...))))))
ψ(1st fixed point of weakly Mahlo after 2nd 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1(1,,2,,)1,,2(1,,3,,)3)2(1,,3,,)3)2}
1757FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1M(2;2)(0)(0)) = ψ(M-I(1,M(2;1)+1))
ψ(1st weakly inaccessible limit of weakly Mahlo after 2nd 2-weakly Mahlo)
{n,n(1(1,,2,,)1,,2(1,,3,,)3)2}
1758FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1M(2;2)(0)) = ψ(M(1,M(2;1)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo after 2nd 2-weakly Mahlo)
{n,n(1(1,,2,,)1(1,,2,,)2(1,,3,,)3)2}
1759FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(2;2)) = ψ(M(1,M(1,M(1,M(1,...M(2;1)+1...)))))
ψ(1st fixed point of weakly Mahlo limit of weakly Mahlo after 2nd 2-weakly Mahlo)
ψ(3d 2-weakly Mahlo)
ψ(χχK(3)(2,4))
{n,n(1(1,,2,,)1(1,,2,,)1,,2(1,,3,,)3)2}
1760FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;ω))
ψ(ω-th 2-weakly Mahlo)
{n,n(1(1,,3,,)1,2)2}
1761FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;ω+1))
ψ(ω+1-th 2-weakly Mahlo)
{n,n(1(1,,3,,)2,2)2}
1762FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;Ω))
ψ(1st uncontable-th 2-weakly Mahlo)
{n,n(1(1,,3,,)1'2)2}
1763FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;M(2;0)))
ψ(2-weakly Mahlo-th 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)2)2)2}
1764FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2M(3;0)(0)(0)(0)(0)) = ψ(ψM(3;0)(0)) = ψ(M(2;M(2;M(2;M(2;...)))))
ψ(1st fixed point of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)1,,2)2)2}
1765FGH ψ-function
ψ(cardinal)
SAN
M(2;ψχχ1χ2M(3;0)(0)(0)(0)(0)+1)
ψ(1st 2-weakly Mahlo after 1st fixed point of 2-weakly Mahlo)
{n,n(1(1,,3,,)2(1(1,,3,,)1,,2)2)2}
1766FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2M(3;0)(0)(0)(0)(1)) = ψ(M(2;M(2;M(2;...M(2;ψχχ1χ2M(3;0)(0)(0)(0)(0)+1)...))))
ψ(2nd fixed point of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)1,,2)3)2}
1767FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2M(3;0)(0)(0)(0)) = ψ(χM(3;0)(0)) = ψ(M(2;α)-I(1,0))
ψ(1st weakly inaccessible limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1,,2)2}
1768FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;χχ1χ2M(3;0)(0)(0)(0)+1)) = ψ(M(2;M(2;α)-I(1,0)+1))
ψ(1st 2-weakly Mahlo after 1st weakly inaccessible limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)2,,2)2}
1769FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2M(3;0)(0)(0)(1)(1)) = ψ(M(2;M(2;M(2;...M(2;χχ1χ2M(3;0)(0)(0)(0)+1)...))))
ψ(1st fixed point of 2-weakly Mahlo after 1st weakly inaccessible limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)1,,2)2,,2)2}
1770FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2M(3;0)(0)(0)(1)) = ψ(M(2;α)-I(1,1))
ψ(2nd weakly inaccessible limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1,,3)2}
1771FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
SAN
ψ(χ1χ2M(3;0)(0)(0)) = ψ(χ1M(3;0)(0)) = ψ(M(2;α)-M(1,0)) = ψ(M(2;α)-I(2,0))
ψ(1st 2-weakly inaccessible limit of 2-weakly Mahlo)
ψ(1st weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1,,1,,2)2}
1772FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1χ2M(3;0)(0)(0)χ1χ2M(3;0)(0)(0)) = ψ(M(2;α)-M(1,0)M(2;α)-M(1,0)) = ψ(M(2;α)-I(1,0,0))
ψ(1st hyper-weakly inaccessible limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,2,,)2)2}
1773FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;χ1χ2M(3;0)(0)(0)+1)) = ψ(M(2;M(2;α)-M(1,0)+1))
ψ(1st 2-weakly Mahlo after 1st weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,2,,)2)2}
1774FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2M(3;0)(0)(1)(1)(1)) = ψ(M(2;M(2;M(2;...M(2;χχ1χ2M(3;0)(0)(0)(0)+1)...))))
ψ(1st fixed point of 2-weakly Mahlo after 1st weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,2,,)2)2}
1775FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2M(3;0)(0)(1)(1)) = ψ(M(2;α)-I(1,M(2;α)-M(1,0)+1))
ψ(1st weakly inaccessible limit of 2-weakly Mahlo after 1st weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)1,,2)2(1,,2,,)2)2}
1776FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1χ2M(3;0)(0)(1)) = ψ(M(2;α)-M(1,1))
ψ(2nd weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1,,1,,2(1,,2,,)2)2}
1777FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
SAN
ψ(χ2M(3;0)(0)) = ψ(M(2;1,0)) = ψ(M(2;α)-M(2,0))
ψ(2nd weakly Mahlo 2-limit of 2-weakly Mahlo)
ψ(1st 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,2,,)1,,2)2}
1778FGH ψ-function
ψ(cardinal)
SAN
ψ(χ2M(3;0)(0)ω) = ψ(M(2;1,0)ω) = ψ(M(2;α)-M(ω,0))
ψ(1st weakly Mahlo ω-limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(2,,2,,)2)2}
1779FGH ψ-function
ψ(cardinal)
SAN
ψ(χ2M(3;0)(0)χ2M(3;0)(0)) = ψ(M(2;1,0)M(2;1,0)) = ψ(M(2;α)-M(1,0,0))
ψ(1st weakly Mahlo hyper-limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,3,,)2)2}
1780FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;χ2M(3;0)(0)+1)) = ψ(M(2;M(2;1,0)))
ψ(1st 2-weakly Mahlo after 1st weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,3,,)2)2}
1781FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2M(3;0)(1)(1)(1)(1)) = ψ(ψM(3;0)(1)) = ψ(M(2;M(2;M(2;...M(2;M(2;1,0)+1)...))))
ψ(1st fixed point of 2-weakly Mahlo after 1st 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1(1,,3,,)1,,2)2(1,,3,,)2)2}
1782FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2M(3;0)(1)(1)(1)) = ψ(χM(3;0)(1)) = ψ(M(2;α)-I(1,M(2;1,0)+1))
ψ(1st weakly inaccessible limit of 2-weakly Mahlo after 1st 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1,,2(1,,3,,)2)2}
1783FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1χ2M(3;0)(1)(1)) = ψ(χ1M(3;0)(1)) = ψ(M(2;α)-M(1,M(2;1,0)+1))
ψ(1st weakly Mahlo limit of 2-weakly Mahlo after 1st 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,2,,)2(1,,3,,)2)2}
1784FGH ψ-function
ψ(cardinal)
SAN
ψ(χ2M(3;0)(1)) = ψ(M(2;1,1))
ψ(2nd 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,3,,)3)2}
1785FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(3;0)) = ψ(M(2;1,M(2;1,M(2;1,M(2;1,...)))))
ψ(1st fixed point of 2-weakly Mahlo limit of 2-weakly Mahlo)
ψ(3-weakly Mahlo)
ψ(χK(3,0)) = ψ(χK(3,3)) = ψ(χK(3,2)) = ψ(χK(3,1)) = ψ(χK(3))
{n,n(1(1,,3,,)1(1,,3,,)1,,2)2}
1786FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;0)+χM(3;0)(M(3;0)) = ψ(M(2;1,α)-I(1,0))
ψ(1st weakly inaccessible limit of 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,3,,)1,,1,,2)2}
1787FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;0)+χ1M(3;0)(M(3;0)) = ψ(M(2;1,α)-M(1,0))
ψ(1st weakly Mahlo limit of 2-weakly Mahlo limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,3,,)1(1,,2,,)2)2}
1788FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;0)+χ2M(3;0)(M(3;0)) = ψ(M(2;2,0))
ψ(1st 2-weakly Mahlo 2-limit of 2-weakly Mahlo)
{n,n(1(1,,3,,)1(1,,3,,)1(1,,3,,)2)2}
1789FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;0)M(3;0)2M(3;0)(M(3;0)M(3;0)) = ψχM(2;1,0,0)(0)(0)
ψ(1st 2-weakly Mahlo hyper-limit of 2-weakly Mahlo)
{n,n(1(1(1(1,,4,,)2)2,,3,,)2)2}
1790FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;0)M(3;0)M(3;0)M(3;0)...) = ψ(εM(3;0)+1) = ψ(ΩM(3;0)+1)
ψ(1st uncountable after 1st 3-weakly Mahlo )
{n,n(1,,2(1,,4,,)2)2}
1791FGH ψ-function
ψ(cardinal)
SAN
ψ(M(1;M(3;0)+1))
ψ(1st weakly Mahlo limit of weakly Mahlo after 1st 3-weakly Mahlo)
{n,n(1(1,,2,,)2(1,,4,,)2)2}
1792FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;M(3;0)+1))
ψ(1st 2-weakly Mahlo limit of weakly Mahlo after 1st 3-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,4,,)2)2}
1793FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(3;1)) = ψ(M(2;1,M(2;1,M(2;1,M(2;1,...M(3;0)+1...)))))
ψ(1st fixed point of 2-weakly Mahlo limit of weakly Mahlo after 1st 3-weakly Mahlo)
ψ(2nd 2-weakly Mahlo)
ψ(χχK(4)(3,4)) = ψ(χK(3,4))
{n,n(1(1,,3,,)1(1,,3,,)1,,2(1,,4,,)2)2}
1794FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(3;2)) = ψ(M(2;1,M(2;1,M(2;1,M(2;1,...M(3;1)+1...)))))
ψ(1st fixed point of 2-weakly Mahlo limit of weakly Mahlo after 2nd 3-weakly Mahlo)
ψ(3d 2-weakly Mahlo)
ψ(χχK(4)(3,5))
{n,n(1(1,,3,,)1(1,,3,,)1,,2(1,,4,,)3)2}
1795FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;ω))
ψ(ω-th 3-weakly Mahlo)
{n,n(1(1,,4,,)1,2)2}
1796FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;M(3;0)))
ψ(3-weakly Mahlo-th 3-weakly Mahlo)
{n,n(1(1,,4,,)1(1(1,,4,,)2)2)2}
1797FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3M(4;0)(0)(0)(0)(0)(0)) = ψ(ψM(4;0)(0)) = ψ(M(3;M(3;M(3;M(3;...)))))
ψ(1st fixed point of 3-weakly Mahlo)
{n,n(1(1,,4,,)1(1(1,,4,,)1,,2)2)2}
1798FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2χ3M(4;0)(0)(0)(0)(0)) = ψ(χM(4;0)(0)) = ψ(M(3;α)-I(1,0))
ψ(1st weakly inaccessible limit of 3-weakly Mahlo)
{n,n(1(1,,4,,)1,,2)2}
1799FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
SAN
ψ(χ1χ2χ3M(4;0)(0)(0)(0)) = ψ(χ1M(4;0)(0)) = ψ(M(3;α)-M(1,0)) = ψ(M(3;α)-I(2,0))
ψ(1st 2-weakly inaccessible limit of 3-weakly Mahlo)
ψ(1st weakly Mahlo limit of 3-weakly Mahlo)
{n,n(1(1,,4,,)1,,1,,2)2}
1800FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
SAN
ψ(χ2χ3M(4;0)(0)(0)) = ψ(χ2M(4;0)(0)) = ψ(M(3;α)-M(2;1,0)) = ψ(M(3;α)-M(2,0))
ψ(1st weakly Mahlo 2-limit of 3-weakly Mahlo)
ψ(1st 2-weakly Mahlo limit of 3-weakly Mahlo)
{n,n(1(1,,4,,)1(1,,2,,)1,,2)2}
1801FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
SAN
ψ(χ3M(4;0)(0)) = ψ(M(3;1,0)) = ψ(M(3;α)-M(2;2,0))
ψ(1st 2-weakly Mahlo 2-limit of 3-weakly Mahlo)
ψ(1st 3-weakly Mahlo limit of 3-weakly Mahlo)
{n,n(1(1,,4,,)1(1,,3,,)1,,2)2}
1802FGH ψ-function
ψ(cardinal)
SAN
ψ(χ3M(4;0)(1)) = ψ(M(3;1,1))
ψ(2nd 3-weakly Mahlo limit of 3-weakly Mahlo)
{n,n(1(1,,4,,)1(1,,4,,)3)2}
1803FGH ψ-function
ψ(cardinal)
ψ(high cardinal)
FGH ψ(ψK)
SAN
ψ(M(4;0)) = ψ(M(3;1,M(3;1,M(3;1,M(3;1,...)))))
ψ(1st fixed point of 3-weakly Mahlo limit of 3-weakly Mahlo)
ψ(1st 4-weakly Mahlo)
ψ(χK(4,0)) = ψ(χK(4,4)) = ψ(χK(4,3)) = ψ(χK(4,2)) = ψ(χK(4,1)) = ψ(χK(4))
{n,n(1(1,,4,,)1(1,,4,,)1,,2)2}
1804FGH ψ-function
ψ(cardinal)
SAN
ψ(M(4;0)M(4;0)3M(4;0)(M(4;0)M(4;0)) = ψχM(3;1,0,0)(0)(0)
ψ(1st 3-weakly Mahlo hyper-limit of 3-weakly Mahlo)
{n,n(1(1(1(1,,4,,)2)2,,3,,)2)2}
1805FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4M(4;0)(0)(0)(0)(0)(0)(0)) = ψ(ψM(5;0)(0)) = ψ(M(4;M(4;M(4;M(4;...)))))
ψ(1st fixed point of 4-weakly Mahlo)
{n,n(1(1,,5,,)1(1(1,,5,,)1,,2)2)2}
1806FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(5;0))
ψ(1st 5-weakly Mahlo)
ψ(χK(5,0)) = ψ(χK(5,5)) = ψ(χK(5,4)) = ψ(χK(5,3)) = ψ(χK(5,2)) = ψ(χK(5,1)) = ψ(χK(5))
{n,n(1(1,,5,,)1(1,,5,,)1,,2)2}
1807FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4...(0)(0)(0)(0)(0)(0)) = ψ(ψM(ω;0)(0)) = ψ(M(sup(n);0))
ψ(1st fixed point of n-weakly Mahlo)
{n,n(1(1,,1,2,,)2)2}
1808FGH ψ-function
ψ(cardinal)
SAN
ψ(ΩψM(ω;0)(0)+1)
ψ(1st uncountable after 1st fixed point of n-weakly Mahlo)
{n,n(1,,2(1,,1,2,,)2)2}
1809FGH ψ-function
ψ(cardinal)
SAN
ψ(MψM(ω;0)(0)+1) = ψ(M(1;ψM(ω;0)(0)+1))
ψ(1st weakly Mahlo after 1st fixed point of n-weakly Mahlo)
{n,n(1(1,,2,,)2(1,,1,2,,)2)2}
1810FGH ψ-function
ψ(cardinal)
SAN
ψ(M(2;ψM(ω;0)(0)+1))
ψ(1st 2-weakly Mahlo after 1st fixed point of n-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,1,2,,)2)2}
1811FGH ψ-function
ψ(cardinal)
SAN
ψ(M(3;ψM(ω;0)(0)+1))
ψ(1st 3-weakly Mahlo after 1st fixed point of n-weakly Mahlo)
{n,n(1(1,,4,,)2(1,,1,2,,)2)2}
1812FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4...(0)(0)(0)(0)(0)(1)) = ψ(M(sup(n);ψM(ω;0)(0)+1))
ψ(2nd fixed point of n-weakly Mahlo)
{n,n(1(1,,1,2,,)3)2}
1813FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2χ3χ4...(0)(0)(0)(0)(0)) = ψ(χM(ω;0)(0)) = ψ(ψχχ1χ2χ3χ4...(0)(0)(0)(0)(0)χχ1χ2χ3χ4...(0)(0)(0)(0)(0)(...)))
ψ(1st weakly inaccessible limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1,,2)2}
1814FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2χ3χ4...(0)(0)(0)(0)(1))
ψ(2nd weakly inaccessible limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1,,3)2}
1815FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1χ2χ3χ4...(0)(0)(0)(0)) = ψ(χ1M(ω;0)(0)) = ψ(χχ1χ2χ3χ4...(0)(0)(0)(0)χ1χ2χ3χ4...(0)(0)(0)(0)(...)))
ψ(1st weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,2,,)2)2}
1816FGH ψ-function
ψ(cardinal)
SAN
ψ(χ2χ3χ4...(0)(0)(0)) = ψ(χ2M(ω;0)(0))
ψ(1st 2-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,3,,)2)2}
1817FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4...(1)(1)(1)(1)(1)(1)) = ψ(ψM(ω;0)(0)) = ψ(χsup(n)M(ω;0)(0))
ψ(1st fixed point of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)2)2}
1818FGH ψ-function
ψ(cardinal)
SAN
ψ(MψM(ω;0)(1)+1) = ψ(M(1;ψM(ω;0)(1)+1))
ψ(1st weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,2,,)2(1,,1,2,,)1(1,,1,2,,)2)2}
1819FGH ψ-function
ψ(cardinal)
SAN
M(2;ψM(ω;0)(1)+1)
ψ(1st 2-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,3,,)2(1,,1,2,,)1(1,,1,2,,)2)2}
1820FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4...(1)(1)(1)(1)(1)(2)) = ψ(M(sup(n);ψM(ω;0)(1)+1))
ψ(2nd fixed point of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)2(1,,1,2,,)2)2}
1821FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1χ2χ3χ4...(1)(1)(1)(1)(1)) = ψ(χM(ω;0)(1)) = ψ(ψχχ1χ2χ3χ4...(1)(1)(1)(1)(1)χχ1χ2χ3χ4...(1)(1)(1)(1)(1)(...)))
ψ(1st weakly inaccessible limit of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1,,2(1,,1,2,,)2)2}
1822FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1χ2χ3χ4...(1)(1)(1)(1)) = ψ(χ1M(ω;0)(1))
ψ(1st weakly Mahlo limit of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,2,,)2(1,,1,2,,)2)2}
1823FGH ψ-function
ψ(cardinal)
SAN
ψ(χ2χ3χ4...(1)(1)(1)) = ψ(χ2M(ω;0)(1))
ψ(1st 2-weakly Mahlo limit of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,3,,)2(1,,1,2,,)2)2}
1824FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1χ2χ3χ4...(2)(2)(2)(2)(2)(2)) = ψ(ψM(ω;0)(2)) = ψ(χsup(n)M(ω;0)(1))
ψ(1st fixed point of n-weakly Mahlo limit of n-weakly Mahlo limit of n-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)3)2}
1825FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω;0)) = ψ(ψM(ω;0)M(ω;0)M(ω;0)(...))))
ψ(1st ω-weakly Mahlo)
ψ(χK(ω,0)) = ψ(χK(ω,n)) = ψ(χK(ω))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,2)2}
1826FGH ψ-function
ψ(cardinal)
SAN
ψ(ψM(ω;1)(0)) = ψ(M(1;M(1;M(1;M(1;...M(ω;0)+1...))))))
ψ(1st fixed point of weakly Mahlo after 1st ω-weakly Mahlo)
{n,n(1(1,,1,2,,)2(1,,1,2,,)1,,2)2}
1827FGH ψ-function
ψ(cardinal)
SAN
ψ(ψM(ω;1)(1))
ψ(1st fixed point of n-weakly Mahlo of weakly Mahlo after 1st ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)2,,2)2}
1828FGH ψ-function
ψ(cardinal)
SAN
ψ(ψM(ω;1)(2))
ψ(1st fixed point of n-weakly Mahlo limit of n-weakly Mahlo of weakly Mahlo after 1st ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)3,,2)2}
1829FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω;1)) = ψ(ψM(ω;1)M(ω;1)M(ω;1)(...))))
ψ(2nd ω-weakly Mahlo)
ψ(χχK(ω+1)(ω,ω+1)) = ψ(χK(ω,ω+1))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,3)2}
1830FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω;2)) = ψ(ψM(ω;2)M(ω;2)M(ω;2)(...))))
ψ(3d ω-weakly Mahlo)
ψ(χχK(ω+1)(ω,ω+2))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,3)2}
1831FGH ψ-function
ψ(cardinal)
SAN
ψ(ψM(ω+1;0)(0)) = ψ(M(ω;M(ω;M(ω;M(ω;...)))))
ψ(1st fixed point of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,1,,2)2}
1832FGH ψ-function
ψ(cardinal)
SAN
ψ(χM(ω+1;0)(0))
ψ(1st weakly inaccessible limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,1,,2)2}
1833FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1M(ω+1;0)(0))
ψ(1st weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,2,,)2)2}
1834FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1...χωM(ω+1)(0)...(1)(1)(1)) = ψ(M(ω;M(ω;M(ω;M(ω;...χsup(n)M(ω+1;0)(0)+1)))))
ψ(1st fixed point of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)2)2}
1835FGH ψ-function
SAN
ψ(Mψχχ1...χωM(ω+1)(0)...(1)(1)(1)+1)
{n,n(1(1,,2,,)2(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)2)2}
1836FGH ψ-function
SAN
ψ(ψM(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1)(1))
{n,n(1(1,,1,2,,)2(1,,1,2,,)1(1,,1,2,,)2)2}
1837FGH ψ-function
SAN
ψ(χM(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1))(1))
{n,n(1(1,,1,2,,)1,,2(1,,1,2,,)1(1,,1,2,,)2)2}
1838FGH ψ-function
SAN
ψ(χ1M(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1))(1))
{n,n(1(1,,1,2,,)1(1,,2,,)2(1,,1,2,,)1(1,,1,2,,)2)2}
1839FGH ψ-function
SAN
ψ(ψM(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1))(2))
{n,n(1(1,,1,2,,)1(1,,1,2,,)2(1,,1,2,,)2)2}
1840FGH ψ-function
SAN
ψ(M(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1)))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,2(1,,1,2,,)2)2}
1841FGH ψ-function
SAN
ψ(M(ω;ψχχ1...χωM(ω+1)(0)...(1)(1)(1)+1))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,3(1,,1,2,,)2)2}
1842FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1...χωM(ω+1)(0)...(1)(1)(2)) = ψ(M(ω;M(ω;M(ω;M(ω;...ψχχ1...χωM(ω+1)(0)...(1)(1)(1)+1)))))
ψ(2nd fixed point of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,1(1(1,,1,2,,)1(1,,1,2,,)1(1,,2,,)2)2(1,,1,2,,)2)2}
1843FGH ψ-function
ψ(cardinal)
SAN
ψ(χχ1...χωM(ω+1)(0)...(1)(1)) = ψ(ψχχ1...χωM(ω+1)(0)...(1)(1)χχ1...χωM(ω+1)(0)...(1)(1)(...)))
ψ(1st weakly inaccessible limit of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,1,,2(1,,1,2,,)2)2}
1844FGH ψ-function
ψ(cardinal)
SAN
ψ(χ1...χωM(ω+1)(0)...(1)) = ψ(χχ1...χωM(ω+1)(0)...(1)χ1...χωM(ω+1)(0)...(1)(...)))
ψ(1st weakly Mahlo limit of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,2,,)2(1,,1,2,,)2)2}
1845FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1...χωM(ω+1)(0)...(2)(2)(2)) = ψ(M(ω;M(ω;M(ω;M(ω;...χsup(n)...χωM(ω+1)(0)...(1)+1)))))
ψ(1st fixed point of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)3)2}
1846FGH ψ-function
ψ(cardinal)
SAN
ψ(ψχχ1...χωM(ω+1)(0)...(3)(3)(3)) = ψ(M(ω;M(ω;M(ω;M(ω;...χsup(n)...χωM(ω+1)(0)...(2)+1)))))
ψ(1st fixed point of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo after 1st fixed point of n-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)4)2}
1847FGH ψ-function
ψ(cardinal)
SAN
ψ(χωM(ω+1;0)(0)) = ψ(χsup(n)...χωM(ω+1)(0)...sup(n)...χωM(ω+1)(0)...(...))) = ψ(M(ω;1,0))
ψ(1st ω-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)1,,2)2}
1848FGH ψ-function
ψ(cardinal)
SAN
ψ(χωM(ω+1;0)(1)) = ψ(M(ω;1,1))
ψ(2nd ω-weakly Mahlo limit of ω-weakly Mahlo)
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)1,,3)2}
1849FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω+1;0)) = ψ(M(ω;1,M(ω;1,M(ω;1,M(ω;1,...)))))
ψ(1st ω+1-weakly Mahlo)
ψ(χK(ω+1,0)) = ψ(χK(ω+1,ω)) = ψ(χK(ω+1,n)) = ψ(χK(ω+1))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)1,,1,,2)2}
1850FGH ψ-function
SAN
ψ(M(ω+1;0)M(ω+1;0)))
{n,n(1(1(1(1,,2,2,,)2)2,,1,2,,)2)2}
1851FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω+2;0)) = ψ(M(ω+1;1,M(ω+1;1,M(ω+1;1,M(ω+1;1,...)))))
ψ(1st ω+2-weakly Mahlo)
ψ(χK(ω+2,0)) = ψ(χK(ω+2,ω+1)) = ψ(χK(ω+2,ω)) = ψ(χK(ω+2,n)) = ψ(χK(ω+2))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)1(1,,1,2,,)1,,1,,2)2}
1852FGH ψ-function
SAN
ψ(ψM(ω×2;0)(0))
{n,n(1(1,,1,3,,)2)2}
1853FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(ω×2;0))
ψ(χK(ω×2,0)) = ψ(χK(ω×2,ω+n)) = ψ(χK(ω×2,n)) = ψ(χK(ω×2))
{n,n(1(1,,1,3,,)1(1,,1,3,,)1,,2)2}
1854FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(M;0))
ψ(1st weakly Mahlo-weakly Mahlo)
ψ(χKK(1)))
{n,n(1(1,,1(1(1,,2,,)2)2,,)1(1,,1(1(1,,2,,)2)2,,)1,,2)2}
1855FGH ψ-function
ψ(cardinal)
FGH ψ(ψK(cardinal))
SAN
ψ(ψM(1,0;0)(0)) = ψ(ψχM(1,0;0)(0)(0)) = ψ(ψχ...M(1,0;0)...(0)(0)) = ψ(M(M(M(...;0);0);0))
ψ(1st fixed point of α-weakly Mahlo)
ψ(K) = ψ(K[1]) = ψ(K[1](1|10;0)) = ψ(ψK(K)) = ψ(ψχK(K)(K)) = ψ(χKKKK(...)))))
ψ(1st Π11-indescribable)
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)2)2} = {n,n(1(1,,1,,2,,)2)2}
1856FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ΩψM(1,0;0)(0)+1)
ψ(1st uncountable after 1st fixed point of α-weakly Mahlo)
ψ(ΩψK(K)+1)
{n,n(1,,2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1857FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(1;ψM(1,0;0)(0)+1))
ψ(1st weakly Mahlo after 1st fixed point of α-weakly Mahlo)
ψ(K+χK(1,K))
{n,n(1(1,,2,,)1(1,,2,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1858FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(2;ψM(1,0;0)(0)+1))
ψ(1st 2-weakly Mahlo after 1st fixed point of α-weakly Mahlo)
ψ(K+χK(2,K))
{n,n(1(1,,3,,)1(1,,3,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1859FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω;ψM(1,0;0)(0)+1))
ψ(1st ω-weakly Mahlo after 1st fixed point of α-weakly Mahlo)
ψ(K+χK(ω,K))
{n,n(1(1,,1,2,,)1(1,,1,2,,)1,,2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1860FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψχM(ψM(1,0;0)(0);0)(0)(1)) = ψ(M(M(M(...M(ω;ψM(1,0;0)(0)+1)...;0);0);0))
ψ(K+ψχKK(K),0)(K+1))
{n,n(1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1861FGH ψ-function
FGH ψ(ψK)
SAN
ψ(χM(ψM(1,0;0)(0);0)(0))
ψ(K+χχKK(K),0)(0,K+1))
{n,n(1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1,,2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1862FGH ψ-function
FGH ψ(ψK)
SAN
ψ(χ1M(ψM(1,0;0)(0);0)(0))
ψ(K+χχKK(K),0)(1,K+1))
{n,n(1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1(1,,2,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1863FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(ψM(1,0;0)(0);0)(1)) = ψ(M(M(M(...M(ω;χψM(1,0;0)(0)M(ψM(1,0;0)(0);0)(0)+1)...;0);0);0))
ψ(K+ψχKK(K),0)(K+2))
{n,n(1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1864FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ψM(1,0;0)(0)+1;0)) = ψ(ψM(ψM(1,0;0)(0);0)M(ψM(1,0;0)(0);0)(...)))
ψ(1st fixed point of α-weakly Mahlo after 1st fixed point of α-weakly Mahlo)
ψ(K+χKK(K),0))
{n,n(1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1,,2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1865FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(M(ψM(1,0;0)(0)+1;0);0))
ψ(1st fixed point of α-weakly Mahlo after 1st fixed point of α-weakly Mahlo after 1st fixed point of α-weakly Mahlo)
ψ(K+χKKK(K),0)))
{n,n(1(1,,1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1(1,,1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)1,,2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1866FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψχM(1,0;0)(0)(1)) = ψ(M(M(...M(ψM(1,0;0)(0)+1;0)...;0);0))
ψ(2nd fixed point of α-weakly Mahlo)
ψ(K+ψχK(K)(K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)3)2}
1867FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψχM(1,0;0)(0)(2)) = ψ(M(M(...M(ψχM(1,0;0)(0)(1)+1;0)...;0);0))
ψ(3d fixed point of α-weakly Mahlo)
ψ(K+ψχχK(0,0)(K)(K+2))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)4)2}
1868FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χM(1,0;0)(0)) = ψ(M(α;0)-I(1,0))
ψ(1st weakly inaccessible limit of α-weakly Mahlo)
ψ(K+χχK(K)(0,K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1,,2)2}
1869FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χ1M(1,0;0)(0)) = ψ(M(α;0)-M(1,0))
ψ(1st weakly Mahlo limit of α-weakly Mahlo )
ψ(K+χχK(K)(1,K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,2,,)2)2}
1870FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χ2M(1,0;0)(0)) = ψ(M(α;0)-M(2;1,0))
ψ(1st 2-weakly Mahlo limit of α-weakly Mahlo )
ψ(K+χχK(K)(2,K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,3,,)2)2}
1871FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χψM(1,0;0)(0)M(1,0;0)(0)) = ψ(M(α;0)-M(M(M(M(M(...;0);0);0);1,0))
ψ(1st fixed point α-weakly Mahlo-weakly Mahlo limit of α-weakly Mahlo )
ψ(K+χχK(K)χK(K)(K),K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2)2}
1872FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χχM(1,0;0)(0)M(1,0;0)(0)) = ψ(M(α;0)-M(M(α;0)-I(1,0);1,0))
ψ(1st weakly inaccessible limit of α-weakly Mahlo-weakly Mahlo limit of α-weakly Mahlo)
ψ(K+χχK(K)χK(K)(0,K+1),K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)1,,2)2,,)2)2}
1873FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χχ1M(1,0;0)(0)M(1,0;0)(0)) = ψ(M(α;0)-M(M(α;0)-M(1,0);1,0))
ψ(1st weakly Mahlo limit of α-weakly Mahlo-weakly Mahlo limit of α-weakly Mahlo)
ψ(K+χχK(K)χK(K)(1,K+1),K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,2,,)2)2,,)2)2}
1874FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χχψM(1,0;0)(0)M(1,0;0)(0)M(1,0;0)(0)) = ψ(M(α;0)-M(M(α;0)-M(M(M(M(...;0);0);0);1,0);1,0))
ψ(1st fixed point of α-weakly Mahlo-weakly Mahlo limit of α-weakly Mahlo-weakly Mahlo limit of α-weakly Mahlo )
ψ(K+χχK(K)χK(K)χK(K)(K),K+1),K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2,,)2)2,,)2)2}
1875FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(1,0;0)(1)) = ψ(ψχM(1,0;0)(1)(1)) = ψ(ψχ...M(1,0;0)...(1)(1)) = ψ(χχχ...M(1,0;0)(0)M(1,0;0)(0)M(1,0;0)(0)) = ψ(M(α;0)-M(M(α;0)-M(M(α;0)-...1,0);1,0);1,0))
ψ(1st fixed point of α-weakly Mahlo limits on α-weakly Mahlo)
ψ(K+ψχK(K)(K+2)) = ψ(K+χχK(K)χK(K)χK(K)(...,K+1),K+1),K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1876FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψχM(1,0;0)(1)(2)) = ψ(M(M(...M(ψM(1,0;0)(1)+1;0)...;0);0))
ψ(1st fixed point of α-weakly Mahlo after 1st fixed point of α-weakly Mahlo limits on α-weakly Mahlo)
ψ(K+ψχχK(0,0)(K)(K+3))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)2(1,,1(1(1,,1,,2,,)2)2,,)2)2}
1877FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(1,0;0)(2)) = ψ(ψχM(1,0;0)(2)(2)) = ψ(ψχ...M(1,0;0)...(2)(2)) = ψ(χχχ...M(1,0;0)(1)M(1,0;0)(1)M(1,0;0)(1))
ψ(2nd fixed point of α-weakly Mahlo limits on α-weakly Mahlo)
ψ(K+ψχK(K)(K+3))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)3)2}
1878FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(1,0;0)M(1,0;0)(0)))
ψ(1st fixed point of α-weakly Mahlo limits on α-weakly Mahlo limits on α-weakly Mahlo)
ψ(K+ψχK(K)(K+ψχK(K)(K)))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)1(1(1,,1(1(1,,1,,2,,)2)2,,)2)2)2}
1879FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(1,0;0)) = ψ(ψM(1,0;0)M(1,0;0)M(1,0;0)(...))))
ψ(1st hyper-weakly Mahlo or 1,0-hyper-weakly Mahlo)
ψ(K+χK(K)) = ψ(K+ψχK(K)(K+ψχK(K)(K+ψχK(K)(K+...))))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)1,,2)2}
1880FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(1,0;1)(0)) = ψ(M(M(...M(1,0;0)+1;0)...;0);0))
ψ(1st fixed point of α-weakly Mahlo after 1st hyper-weakly Mahlo)
ψ(K+ψχK(K+1)(K,K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)2(1,,1(1(1,,1,,2,,)2)2,,)1,,2)2}
1881FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(1,0;1)) = ψ(ψM(1,0;1)M(1,0;1)M(1,0;1)(...))))
ψ(2nd hyper-weakly Mahlo or 1,0-hyper-weakly Mahlo)
ψ(K+χχK(K+1)(K,K+1)) = ψ(K+χK(K,K+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)2,,)1,,3)2}
1882FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(1,1;0)(0)) = ψ(M(1,0;M(1,0;M(1,0;...);0);0);0))
ψ(1st fixed point of hyper-weakly Mahlo or 1,0-hyper-weakly Mahlo)
ψ(K+ψχK(K+1)(K+1))
{n,n(1(1,,2(1(1,,1,,2,,)2)2,,)2)2}
1883FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(1,1;0)) = ψ(ψM(1,1;0)M(1,1;0)M(1,1;0)(...))))
ψ(1st 2-hyper-weakly Mahlo or 1,0-hyper-weakly Mahlo)
ψ(K+χK(K+1))
{n,n(1(1,,2(1(1,,1,,2,,)2)2,,)1(1,,2(1(1,,1,,2,,)2)2,,)1,,2)2}
1884FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(2,0;0)(0)) = ψ(M(M(...M(1,0;0)+1;0)...;0);0))
ψ(1st fixed point of α-hyper-weakly Mahlo or 1,0-hyper-weakly Mahlo)
ψ(K×2) = ψ(K+K) = ψ(K+χK(K+χK(K+χK(K+...))))
{n,n(1(1,,1(1(1,,1,,2,,)2)3,,)2)2}
1885FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(2,0;0)) = ψ(M(1,M(1,M(1,M(1,...;0);0);0);0))
ψ(1st hyper-hyper-weakly Mahlo or 2,0-hyper-weakly Mahlo)
ψK(K×2+χK(K×2))
{n,n(1(1,,1(1(1,,1,,2,,)2)3,,)1(1,,1(1(1,,1,,2,,)2)3,,)1,,2)2}
1886FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(ω,0;0)(0))
ψ(1st fixed point of α-hypern-weakly Mahlo)
ψ(K×ω)
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)2)2}
1887FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψχM(ω,0;0)(0)(1))
ψ(2nd fixed point of α-hypern-weakly Mahlo)
ψ(K×ω+ψχK(K×ω)(K×ω+1))
{n,n(1(1,,1(1(1,,1,,2,,)1,2)2,,)3)2}
1888FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χM(2,0;0)(0))
ψ(1st weakly inaccessible limit of α-hypern-weakly Mahlo)
ψ(K×ω+χχK(K×ω,0)(0,K×ω+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)1,,2)2}
1889FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χ1M(2,0;0)(0))
ψ(1st weakly inaccessible limit of α-hypern-weakly Mahlo)
ψ(K×ω+χχK(K×ω)(1,K×ω+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)1(1,,2,,)2)2}
1890FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(χ2M(2,0;0)(0))
ψ(1st weakly inaccessible limit of α-hypern-weakly Mahlo)
ψ(K×ω+χχK(K×ω)(2,K×ω+1))
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)1(1,,3,,)2)2}
1891FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(ψM(ω,0;0)(1))
ψ(1st fixed point of α-hypern-weakly Mahlo limits on α-hypern-weakly Mahlo)
ψ(K×ω+ψχK(K×ω)(K×ω+2))
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)1(1,,1(1(1,,1,,2,,)2)1,2,,)2)2}
1892FGH ψ-function
ψ(cardinal)
FGH ψ(ψK)
SAN
ψ(M(ω,0;0))
ψ(1st α-hyperω-weakly Mahlo)
ψ(K×ω+χK(K×ω))
{n,n(1(1,,1(1(1,,1,,2,,)2)1,2,,)1(1,,1(1(1,,1,,2,,)2)1,2,,)1,,2)2}
1893FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1,0,0;0)(0)) = ψ(M(M(M(M(...,0;0),0;0),0;0),0;0))
ψ(K2)
{n,n(1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)2)2}
1894FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0;0))
ψ(K2K(K2))
{n,n(1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)1,,2)2}
1895FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1,0,0,0;0)(0))
ψ(K3)
{n,n(1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)2)2}
1896FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0,0;0))
ψ(K3K(K3))
{n,n(1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)1,,2)2}
1897FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1,0,0,0,...;0)(0)) = ψ(ψM(1ω;0)(0))
ψ(Kω)
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)2)2}
1898FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1;ψM(1,0,0,0,...;0)(0)+1))
ψ(KωK(1,Kω))
{n,n(1(1,,2,,)2(1,,1(2(1,,1,,2,,)2)2,,)2)2}
1899FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(2;ψM(1,0,0,0,...;0)(0)+1))
ψ(KωK(2,Kω))
{n,n(1(1,,3,,)2(1,,1(2(1,,1,,2,,)2)2,,)2)2}
1900FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψχM(1,0,0,...;0)(0)(1)) = ψ(ψM(1,0,0,0,...ψM(1,0,0,0,0,...;0)(0)+1;0)(0))
ψ(KωχK(Kω)(Kω+1))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)3)2}
1901FGH ψ-function
FGH ψ(ψK)
SAN
ψ(χM(1,0,0,0,...;0)(0))
ψ(KωχK(Kω)(0,Kω+1))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1,,2)2}
1902FGH ψ-function
FGH ψ(ψK)
SAN
ψ(χ1M(1,0,0,0,...;0)(0))
ψ(KωχK(Kω)(1,Kω+1))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,2,,)2)2}
1903FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1,0,0,0,...;0)(1))
ψ(KωχK(Kω)(Kω+2))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,1(2(1,,1,,2,,)2)2,,)2)2}
1904FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0,0,...;0)) = ψ(M(1ω;0))
ψ(KωK(Kω))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,1(2(1,,1,,2,,)2)2,,)1,,2)2}
1905FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0,0,...;1))
ψ(KωχK(Kω+1,0)(Kω,Kω+1)) = ψ(KωK(Kω,Kω+1))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,1(2(1,,1,,2,,)2)2,,)1,,3)2}
1906FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0,0,...;M(1,0,0,0,...;0)))
ψ(KωχK(Kω+1,0)(Kω,KωK(Kω)))
{n,n(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,1(2(1,,1,,2,,)2)2,,)1,,1(1(1,,1(2(1,,1,,2,,)2)2,,)1(1,,1(2(1,,1,,2,,)2)2,,)1,,2)2)2}
1907FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1,0,0,0,...1;0)) = ψ(M(1ω10;0))
ψ(KωK(Kω+1))
{n,n(1(1,,2(2(1,,1,,2,,)2)2,,)2)2}
1908FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(2,0,0,0,...;0)) = ψ(ψM(2ω;0)(0))
ψ(Kω×2)
{n,n(1(1,,1(2(1,,1,,2,,)2)3,,)2)2}
1909FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(ω,0,0,0,...;0)) = ψ(ψM(2ω;0)(0))
ψ(Kω×ω)
{n,n(1(1,,1(1,2(1,,1,,2,,)2)3,,)2)2}
1910FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1ω+1;0)(0))
ψ(Kω+1)
{n,n(1(1,,1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2,,)2)2}
1911FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1ω×2;0)(0))
ψ(Kω×2)
{n,n(1(1,,1(2(1,,1,,2,,)2)1(2(1,,1,,2,,)2)2,,)2)2}
1912FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1ω2;0)(0))
ψ(Kω2)
{n,n(1(1,,1(3(1,,1,,2,,)2)2,,)2)2}
1913FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1M(1M(1M(1...;0);0);0);0)(0))
ψ(KK)
{n,n(1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)2)2}
1914FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψχM(1M(1M(1M(1...;0);0);0);0)(0)(1))
ψ(KKχK(KK,0)(KK+1))
{n,n(1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)3)2}
1915FGH ψ-function
FGH ψ(ψK)
SAN
ψ(ψM(1M(1M(1M(1...;0);0);0);0)(1))
ψ(KKχK(KK,0)(KK+2))
{n,n(1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)2)2}
1916FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1M(1M(1M(1...;0);0);0);0))
ψ(KKK(KK))
{n,n(1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)1,,2)2}
1917FGH ψ-function
FGH ψ(ψK)
SAN
ψ(M(1M(1M(1M(1...;1);1);1);1))
ψ(KKK(KK+1))
{n,n(1(1,,2(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)2)2}
1918FGH ψ-function
SAN
ψ(KK×2)
{n,n(1(1,,1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)3,,)2)2}
1919FGH ψ-function
SAN
ψ(KK×ω)
{n,n(1(1,,1(2(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2,,)2)2}
1920FGH ψ-function
SAN
ψ(KK2)
{n,n(1(1(1,,1,,2,,)2)3(1,,1,,2,,)2)2}
1921FGH ψ-function
SAN
ψ(KKK)
{n,n(1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1922FGH ψ-function
SAN
ψ(KKKK)
{n,n(1(1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1923FGH ψ-function
SAN
ψ(KKKKK)
{n,n(1(1(1(1,,1,,2,,)2)1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1924name
proof theory
FGH ψ-function

ψ(cardinal)
SAN
Large Rathjen ordinal
KP+П3
ψ(KKKK...) = ψ(εK+1) = ψ(ΩK+1)
ψ(1st uncountable after 1st Π11-indescribable)
{n,n(1,,2(1,,1,,2,,)2)2}
1925FGH ψ-function
ψ(cardinal)
SAN
ψ(ΩK+2)
ψ(2nd uncountable after 1st Π11-indescribable)
{n,n(1,,3(1,,1,,2,,)2)2}
1926FGH ψ-function
SAN
ψ(ΩK×2)
{n,n(1,,1(1(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1927FGH ψ-function
SAN
ψ(ΩΩK+1)
{n,n(1,,1(1,,2(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1928FGH ψ-function
SAN
ψ(ΩΩΩK+1)
{n,n(1,,1(1,,1(1,,2(1,,1,,2,,)2)2(1,,1,,2,,)2)2(1,,1,,2,,)2)2}
1929FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(ΩΩΩ...ΩK+1)
ψ(1st fixed point of uncountable after 1st Π11-indescribable)
ψ(IK+1)
ψ(1st weakly inaccessible after 1st Π11-indescribable)
{n,n(1,,1,,2(1,,1,,2,,)2)2}
1930FGH ψ-function
ψ(cardinal)

SAN
ψ(IK+2)
ψ(2nd weakly inaccessible after 1st Π11-indescribable)
{n,n(1,,1,,3(1,,1,,2,,)2)2}
1931FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(III...IK+1)
ψ(1st fixed point of weakly inaccessible after 1st Π11-indescribable)
ψ(I(2,K+1))
ψ(2-weakly inaccessible after 1st Π11-indescribable)
{n,n(1,,1,,1,,2(1,,1,,2,,)2)2}
1932FGH ψ-function
ψ(cardinal)

SAN
ψ(I(3,K+1))
ψ(3-weakly inaccessible after 1st Π11-indescribable)
{n,n(1,,1,,1,,1,,2(1,,1,,2,,)2)2}
1933FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(I(I(I(I(K+1,1),...0),0),0))
ψ(1st fixed point of α-weakly inaccessible after 1st Π11-indescribable)
ψ(MK+1)
ψ(1st weakly Mahlo after 1st Π11-indescribable)
{n,n(1(1,,2,,)2(1,,1,,2,,)2)2}
1934FGH ψ-function
ψ(cardinal)

SAN
ψ(MK+2)
ψ(2nd weakly Mahlo after 1st Π11-indescribable)
{n,n(1(1,,2,,)3(1,,1,,2,,)2)2}
1935FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(MMM...MK+1)
ψ(1st fixed point of weakly Mahlo after 1st Π11-indescribable)
ψ(M(1,K+1))
ψ(1st weakly Mahlo limit after 1st Π11-indescribable)
{n,n(1(1,,2,,)1(1,,2,,)2(1,,1,,2,,)2)2}
1936FGH ψ-function
ψ(cardinal)

SAN
ψ(M(2,K+1))
ψ(2nd weakly Mahlo limit after 1st Π11-indescribable)
{n,n(1(1,,2,,)1(1,,2,,)1(1,,2,,)2(1,,1,,2,,)2)2}
1937FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(M(M(M(M(K,1),...0),0),0))
ψ(1st fixed point of weakly Mahlo α-limit after 1st Π11-indescribable)
ψ(M(2;K+1))
ψ(1st 2-weakly Mahlo after 1st Π11-indescribable)
{n,n(1(1,,3,,)2(1,,1,,2,,)2)2}
1938FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(M(M(M(M(K+1;0);...0);0);0))
ψ(1st fixed point of α-weakly Mahlo after 1st Π11-indescribable)
ψ(K2)
ψ(2nd Π11-indescribable)
{n,n(1(1,,1,,2,,)3)2}
1939FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(KKKK...)
ψ(1st fixed point of Π11-indescribable)
ψ(K-I(1,0))
ψ(1st weakly inaccessible limit of Π11-indescribable)
{n,n(1(1,,1,,2,,)1,,2)2}
1940FGH ψ-function
ψ(cardinal)

SAN
ψ(K-I(2,0))
ψ(1st 2-weakly inaccessible limit of Π11-indescribable)
{n,n(1(1,,1,,2,,)1,,1,,2)2}
1941FGH ψ-function
ψ(cardinal)

SAN
ψ(K-M(1,0))
ψ(1st weakly Mahlo limit of Π11-indescribable)
{n,n(1(1,,1,,2,,)1(1,,2,,)2)2}
1942FGH ψ-function
ψ(cardinal)

SAN
ψ(K-M(2;1,0))
ψ(1st 2-weakly Mahlo limit of Π11-indescribable)
{n,n(1(1,,1,,2,,)1(1,,3,,)2)2}
1943FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1,0))
ψ(1st Π11-indescribable limit of Π11-indescribable)
{n,n(1(1,,1,,2,,)1(1,,1,,2,,)2)2}
1944FGH ψ-function
ψ(cardinal)

SAN
ψ(K~M(1;0))
ψ(1st weakly Mahlo in which Π11-indescribable are stationary)
{n,n(1(1,,2,,2,,)2)2}
1945FGH ψ-function
ψ(cardinal)

SAN
ψ(K~M(2;0))
ψ(1st weakly Mahlo in which (weakly Mahlos in which Π11-indescribable are stationary) are stationary or 1st 2-weakly Mahlo in which Π11-indescribable are stationary)
{n,n(1(1,,3,,2,,)2)2}
1946FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;0))
ψ(1st Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)2)2}
1947FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;1))
ψ(2nd Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)3)2}
1948FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)-I(1,0))
ψ(1st weakly inaccessible limit of Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)1,,2)2}
1949FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)-M(1,0))
ψ(1st weakly Mahlo limit of Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)1(1,,2,,)2)2}
1950FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)-M(2;1,0))
ψ(1st 2-weakly Mahlo limit of Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)1(1,,3,,)2)2}
1951FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)-K(1,0))
ψ(1st Π11-indescribable limit of Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)1(1,,1,,2,,)2)2}
1952FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;1,0))
ψ(1st Π11-indescribable in which Π11-indescribable are stationary limit of Π11-indescribable in which Π11-indescribable are stationary)
{n,n(1(1,,1,,3,,)1(1,,1,,3,,)2)2}
1953FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)~M(1;0))
ψ(1st weakly Mahlo in which (Π11-indescribable in which Π11-indescribable are stationary) are stationary)
{n,n(1(1,,2,,3,,)2)2}
1954FGH ψ-function
ψ(cardinal)

SAN
ψ(K(1;α)~M(2;0))
ψ(1st 2-weakly Mahlo in which (Π11-indescribable in which Π11-indescribable are stationary) are stationary)
{n,n(1(1,,3,,3,,)2)2}
1955FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;0)) = ψ(K(1;α)~K(1;0))
ψ(1st Π11-indescribable in which (Π11-indescribable in which Π11-indescribable are stationary) are stationary or 1st Π11-indescribable in which Π11-indescribable are 2-stationary)
{n,n(1(1,,1,,4,,)2)2}
1956FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;1))
ψ(2nd Π11-indescribable in which Π11-indescribable are 2-stationary)
{n,n(1(1,,1,,4,,)3)2}
1957FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;α))-K(1,0))
ψ(1st Π11-indescribable limit of Π11-indescribable in which Π11-indescribable are 2-stationary)
{n,n(1(1,,1,,4,,)1(1,,1,,2,,)2)2}
1958FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;α)-K(1;1,0))
ψ(1st Π11-indescribable in which Π11-indescribable are stationary limit of Π11-indescribable in which Π11-indescribable are 2-stationary)
{n,n(1(1,,1,,4,,)1(1,,1,,3,,)2)2}
1959FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;1,0))
ψ(1st Π11-indescribable in which Π11-indescribable are 2-stationary limit of Π11-indescribable in which Π11-indescribable are 2-stationary)
{n,n(1(1,,1,,4,,)1(1,,1,,4,,)2)2}
1960FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;α)~M(1;0))
ψ(1st weakly Mahlo in which (Π11-indescribable in which Π11-indescribable are 2-stationary) are stationary)
{n,n(1(1,,2,,4,,)2)2}
1961FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2;α)~M(2;0))
ψ(1st 2-weakly Mahlo in which (Π11-indescribable in which Π11-indescribable are 2-stationary) are stationary)
{n,n(1(1,,3,,4,,)2)2}
1962FGH ψ-function
ψ(cardinal)

SAN
ψ(K(3;0)) = ψ(K(2;α)~K(1;0))
ψ(1st Π11-indescribable in which (Π11-indescribable in which (Π11-indescribable in which Π11-indescribable are stationary) are stationary) are stationary or 1st Π11-indescribable in which Π11-indescribable are 3-stationary)
{n,n(1(1,,1,,5,,)2)2}
1963FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(K(1,0;0))
ψ(1st Π11-indescribable in which Π11-indescribable are hyper-stationary)
ψ(K(2|10;0))
ψ(1st Π11-indescribable with Π11-indescribablity over Π11-indescribable = 1st Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)2)2}
1964FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;1))
ψ(2nd Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)3)2}
1965FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-I(1,0))
ψ(1st weakly inaccessible limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1,,2)2}
1966FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-M(1,0))
ψ(1st weakly Mahlo limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,2,,)2)2}
1967FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-M(2;1,0))
ψ(1st 2-weakly Mahlo limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,3,,)2)2}
1968FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-K(1,0))
ψ(1st Π11-indescribable limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,1,,2,,)2)2}
1969FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-K(1;1,0))
ψ(1st Π11-indescribable in which Π11-indescribable are stationary limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,1,,3,,)2)2}
1970FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)-K(2;1,0))
ψ(1st Π11-indescribable in which Π11-indescribable are 2-stationary limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,1,,4,,)2)2}
1971FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;1,0))
ψ(1st Π11-indescribable limit of Π11-2-indescribable)
{n,n(1(1,,1,,1,,2,,)1(1,,1,,1,,2,,)2)2}
1972FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)~M(1;0))
ψ(1st weakly Mahlo in which Π11-2-indescribable are stationary)
{n,n(1(1,,2,,1,,2,,)2)2}
1973FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)~M(2;0))
ψ(1st 2-weakly Mahlo in which Π11-2-indescribable are stationary)
{n,n(1(1,,3,,1,,2,,)2)2}
1974FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)~K(1;0))
ψ(1st Π11-indescribable in which Π11-2-indescribable are stationary)
{n,n(1(1,,1,,2,,2,,)2)2}
1975FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|10;α)~K(2;0))
ψ(1st Π11-indescribable in which (Π11-indescribable in which Π11-2-indescribable are stationary) are 2-stationary)
{n,n(1(1,,1,,3,,2,,)2)2}
1976FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|11;0))
ψ(1st Π11-2-indescribable in which Π11-2-indescribable are stationary)
{n,n(1(1,,1,,1,,3,,)2)2}
1977FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|11;1))
ψ(2nd Π11-2-indescribable in which Π11-2-indescribable are stationary)
{n,n(1(1,,1,,1,,3,,)3)2}
1978FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|11;1,0))
ψ(1st Π11-2-indescribable in which Π11-2-indescribable are stationary limit of Π11-2-indescribable in which Π11-2-indescribable are stationary)
{n,n(1(1,,1,,1,,3,,)1,,2)2}
1979FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|12;0))
ψ(1st Π11-2-indescribable in which (Π11-2-indescribable in which Π11-2-indescribable are stationary) are stationary or 1st Π11-2-indescribable in which Π11-2-indescribable are 2-stationary)
{n,n(1(1,,1,,1,,3,,)2)2}
1980FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|12;1))
ψ(2nd Π11-2-indescribable in which Π11-2-indescribable are 2-stationary)
{n,n(1(1,,1,,1,,4,,)3)2}
1981FGH ψ-function
ψ(cardinal)

SAN
ψ(K(2|12;1,0))
ψ(1st Π11-2-indescribable in which Π11-2-indescribable are 2-stationary limit of Π11-2-indescribable in which Π11-2-indescribable are 2-stationary)
{n,n(1(1,,1,,1,,4,,)1,,2)2}
1982FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(K(2|11,0;0))
ψ(1st Π11-2-indescribable in which Π11-2-indescribable are hyper-stationary)
ψ(K(3|10;0))
ψ(1st Π11-indescribable with Π11-indescribablity over Π11-indescribable with Π11-indescribablity over Π11-indescribable = 1st Π11-3-indescribable)
{n,n(1(1,,1,,1,,1,,2,,)2)2}
1983FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(K(1,0|10;0))
ψ(1st Π11-hyper-indescribable)
ψ(K[2]) = ψ(K[2](1|20|10;0))
ψ(1st Π12-indescribable)
{n,n(1(1(1,,2,,)2,,)2)2}
1984name
proof theory
FGH ψ-function

ψ(cardinal)
SAN
Duchhardt ordinal
KP+П4
ψ(K[2]K[2]K[2]K[2]...) = ψ(εK[2]+1) = ψ(ΩK[2]+1)
ψ(1st uncountable after 1st Π12-indescribable)
{n,n(1,,2(1(1,,2,,)2,,)2)2}
1985FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]-I(1,0))
ψ(1st weakly inaccessible limit of Π12-indescribable)
{n,n(1(1(1,,2,,)2,,)1,,2)2}
1986FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]-M(1,0))
ψ(1st weakly Mahlo limit of Π12-indescribable)
{n,n(1(1(1,,2,,)2,,)1(1,,2,,)2)2}
1987FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]-K(1,0))
ψ(1st Π11-indescribable limit of Π12-indescribable)
{n,n(1(1(1,,2,,)2,,)1(1,,1,,2,,)2)2}
1988FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2](1|20|10;1,0))
ψ(1st Π12-indescribable limit of Π12-indescribable)
{n,n(1(1(1,,2,,)2,,)1(1(1,,2,,)2,,)2)2}
1989FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]~M(1;0))
ψ(1st weakly Mahlo in which Π12-indescribable are stationary)
{n,n(1(1,,2(1,,2,,)2,,)2)2}
1990FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]~K(1;0))
ψ(1st Π11-indescribable in which Π12-indescribable are stationary)
{n,n(1(1,,1,,2(1,,2,,)2,,)2)2}
1991FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2](1|20|11;0))
ψ(1st Π12-indescribable in which Π12-indescribable are stationary)
{n,n(1(1,,1(1,,2,,)3,,)2)2}
1992FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2]-1-K(1|10))
ψ(1st Π11-indescribable with Π11-indescribablity over Π12-indescribable)
{n,n(1(1,,1(1,,2,,)1,,2,,)2)2}
1993FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2](1|21|10;0))
ψ(1st Π12-indescribable with Π11-indescribablity over Π12-indescribable)
{n,n(1(1,,1(1,,2,,)1(1,,2,,)2,,)2)2}
1994FGH ψ-function
ψ(cardinal)

SAN
ψ(K[2](2|20|10;0))
ψ(1st Π12-indescribable with Π12-indescribablity over Π12-indescribable = 1st Π12-2-indescribable)
{n,n(1(1,,1(1,,3,,)2,,)2)2}
1995FGH ψ-function
FGH ψ(higher
)
ψ(cardinal)

ψ(high cardinal)
SAN
ψ(K[2](1,0|20|10;0))
ψ(1st Π12-hyper-indescribable)
ψ(K[3])
ψ(1st Π13-indescribable)
{n,n(1(1(1,,1,,2,,)2,,)2)2}
1996proof theory
FGH ψ-function

ψ(cardinal)
SAN
KP+П5
ψ(K[3]K[3]K[3]K[3]...) = ψ(εK[3]+1) = ψ(ΩK[3]+1)
ψ(1st uncountable after 1st Π13-indescribable)
{n,n(1,,2(1(1,,1,,2,,)2,,)2)2}
1997FGH ψ-function
ψ(cardinal)

SAN
ψ(K[4])
ψ(1st Π14-indescribable)
{n,n(1(1(1(1,,2,,)2,,)2,,)2)2}
1998proof theory
FGH ψ-function

ψ(cardinal)
SAN
KP+П6
ψ(K[4]K[4]K[4]K[4]...) = ψ(εK[4]+1) = ψ(ΩK[4]+1)
ψ(1st uncountable after 1st Π14-indescribable)
{n,n(1,,2(1(1(1,,2,,)2,,)2,,)2)2}
1999FGH ψ-function
ψ(cardinal)
SAN (even)
SAN (odd)
ψ(K[n])
ψ(1st Π1n-indescribable)
{n,n(1(1,,2,,)2)2} ~n/2-1 insert
{n,n(1(1,,1,,2,,)2)2} ~(n-1)/2-1 insert
2000proof theory
FGH ψ-function

ψ(cardinal)
SAN (even)
SAN (odd)
KP+Пn+2
ψ(K[n]K[n]K[n]K[n]...) = ψ(εK[n]+1) = ψ(ΩK[n]+1)
ψ(1st uncountable after 1st Π1n-indescribable)
{n,n(1,,2(1,,2,,)2)2} ~n/2-1 insert
{n,n(1,,2(1,,1,,2,,)2)2} ~(n-1)/2-1 insert

few more jumps

1 symbol
count.ordinal
cardinal equiv.
collapse
Ω
ω1CK = 1st admissible = П11-comprehension
ω1 = 1st uncountable = 1st 1-regular
ψ(εΩ+1) - Bachmann-Howard ordinal - PTO of KP
2 symbol
count.ordinal
cardinal equiv.
Ω2
ω2CK = 2nd admissible
ω2 = 2nd uncountable = 2nd 1-regular
3 symbol
count.ordinal
cardinal equiv.
collapse
Ωω
ωωCK = ω-th 1-admissible
ωω = ω-th uncountable = ω-th 1-regular
ψ(Ωω) - Buchholz ordinal - PTO of П11-CA0
4 symbol
count.ordinal
cardinal equiv.
collapse
ΩΩΩΩ...
1st fixed point admissible
1st fixed point uncountable
ψ(ΩΩΩΩ...) - PTO of П11-TR0
5 symbol
count.ordinal
cardinal equiv.
collapse
I
recursively inaccessible 1-admissible
weakly inaccessible 
ψ(εI+1) - Small Rathjen ordinal - PTO of KPI
6 symbol
count.ordinal
cardinal equiv.
collapse
M
recursively Mahlo 1-admissible
weakly Mahlo 
ψ(εM+1) - Rathjen ordinal - PTO of KPM
7 symbol
count.ordinal
cardinal equiv.
collapse
K = K[1]
П3-reflection = 2-admissible
П11-indescribable = 2-regular = weakly copmact
ψ(εK+1) - Large Rathjen ordinal - PTO of KP+П3
8 symbol
count.ordinal
cardinal equiv.
collapse
K[2]
П4-reflection = 3-admissible
П12-indescribable = 3-regular
ψ(εK[2]+1) - Duchhardt ordinal - PTO of KP+П4
9 symbol
count.ordinal
cardinal equiv.
collapse
K[n]
Пn+2-reflection = n+1-admissible
П1n-indescribable = n+1-regular
ψ(εK[n]+1) - PTO of KP+Пn
10 symbol
count.ordinal
cardinal equiv.
collapse
K[ω] = Ξ
Пω-reflection = П10-reflection = ω-admissible = (+1)-stable
1-П10-indescribable =  П1ω-indescribable = П20-indescribable = ω-regular
ψ(εΞ+1) - Small Stegert ordinal - PTO of KP+Пω
11 symbol
count.ordinal
cardinal equiv.
K[ω+1]
П11-reflection = ω+1-admissible
1-П11-indescribable =  П21-indescribable = ω+1-regular
12 symbol
count.ordinal
cardinal equiv.
K[ω+2]
П12-reflection = ω+2-admissible
1-П12-indescribable =  П22-indescribable = ω+2-regular
13 symbol
count.ordinal
cardinal equiv.
K[ω×2]
(+2)-stable =ω×2-admissible
2-П1ω-indescribable = П30-indescribable = ω×2-regular
14 symbol
count.ordinal
cardinal equiv.
K[ω×n]
(+n)-stable = ω×n-admissible
n-П1ω-indescribable = Пnm-indescribable = ω×n-regular
15 symbol
count.ordinal
cardinal equiv.
K[ω2]
(+ω)-stable ordinal = ω2-admissible
ω-П10-indescribable = ω2-regular
16 symbol
count.ordinal
cardinal equiv.
K[ε0]
(+ε0)-stable = ε0-admissible)
ε010-indescribable = ε0-regular
17 symbol
count.ordinal
cardinal equiv.
K[Ω]
(+)-stable = 1st 1-admissible-admissible ordinal
ω110-indescribable
18 symbol
count.ordinal
cardinal equiv.
K[Ω2]
(++)-stable = 2nd 1-admissible-admissible ordinal
ω210-indescribable
19 symbol
count.ordinal
cardinal equiv.
K[I]
recursively inaccessible 1-admissible-admissible ordinal = inaccessibly-stable
I-П10-indescribable
20 symbol
count.ordinal
cardinal equiv.
K[M]
recursively Mahlo 1-admissible-admissible ordinal = Mahlo-stable
M-П10-indescribable
21 symbol
count.ordinal
cardinal equiv.
K[K[1]]
2-admissible-stable = 2-admissible-admissible
П11-indescribable-П10-indescribable
22 symbol
count.ordinal
cardinal equiv.
K[K[ω]]
double (+1)-stable ordinal = ω-admissible-admissible
1-П10-indescribable-П10-indescribable
23 symbol
count.ordinal
cardinal equiv.
K[K[K[ω]]]
triple (+1)-stable = ω-admissible-admissible-admissible
1-П10-indescribable-П10-indescribable-П10-indescribable
24 symbol
count.ordinal
cardinal equiv.
collapse
K[α] = K[K[K[K[K[...]]]]] = ϒ
ω-le (+1)-stable ordinal  = α-admissible
subtle
ψ(εϒ+1) - Large Stegert ordinal - PTO of KP+stable
25 symbol
count.ordinal
K[α]-I(1,0) = K[α]→I
inaccessible limit of α-admissible = inaccessible limit of α-le (+1)-stable ordinal
26 symbol
count.ordinal
K[α]~M(1;0) = K[α]→M
1st Mahlo in which α-admissible are stationary
27 symbol
count.ordinal
K[α]-1-K(1|10)) = K[α]→K
2-admissible with П3-reflection over α-admissible
28 symbol
count.ordinal
K[α]-2-K[2](1|20)) = K[α]→K[2]
3-admissible with П4-reflection over α-admissible
29 symbol
count.ordinal
K[α]→K[ω]
ω-admissible with Пω-reflection over α-admissible
30 symbol
count.ordinal
K[α]→K[α]→K[α]→K[α]→...
fixed point-α-admissible
31 symbol
count.ordinal
cardinal equiv.
Ω{1}
unbounded in any stable = 1st nonprojectible = 1st strong admissible = 1st S1-admissible = П12-comprehension
2-ineffable
ψ(εΩ{1}+1) - PTO of KP+unbounded stable
32 symbol
count.ordinal
Ω2{1}
2nd nonprojectible = 2nd strong admissible = 2nd S1-admissible
33 symbol
count.ordinal
collapse
Ωω{1}
ω-th nonprojectible = ω-th strong admissible = ω-th S1-admissible
ψ(Ωω{1}) - PTO of П12-CA0
34 symbol
count.ordinal
collapse
ΩΩΩΩ...{1}{1}{1}{1}
1st fixed point of nonprojectible = 1st fixed point of strong admissible = 1st fixed point of S1-admissible
ψ(ΩΩΩΩ...{1}{1}{1}{1}) - PTO of П12-TR0
35 symbol
count.ordinal
I{1}
recursively inaccessible nonprojectible = recursively inaccessible strong admissible = recursively inaccessible S1-admissible
36 symbol
count.ordinal
M{1}
recursively Mahlo nonprojectible = recursively inaccessible strong admissible = recursively inaccessible S1-admissible
37 symbol
count.ordinal
K{1} = K{1}[1]
strong 2-admissible = 2-S1-admissible
38 symbol
count.ordinal
K{1}[2]
strong 3-admissible = 3-S1-admissible
39 symbol
count.ordinal
K{1}[ω]
strong ω-admissible = ω-S1-admissible = (+1)-2-stable
40 symbol
count.ordinal

cardinal equiv.
Ω{2}
unbounded in any 2-stable = 1st 2-strong admissible = 1st S2-admissible = П13-comprehension
3-ineffable
41 symbol
count.ordinal
collapse
Ωω{2}
ω-th 2-strong admissible = ω-th S2-admissible
ψ(Ωω{2}) - PTO of П13-CA0
42 symbol
count.ordinal
collapse
ΩΩΩΩ...{2}{2}{2}{2}
1st fixed point of S2-admissible
ψ(ΩΩΩΩ...{2}{2}{2}{2}) - PTO of П13-TR0
43 symbol
count.ordinal
K{2}[ω]
ω-admissible = ω-S2-admissible = (+1)-3-stable
44 symbol
count.ordinal

cardinal equiv.
Ω{3}
unbounded in any 3-stable = 1st S3-admissible = П14-comprehension
4-ineffable
45 symbol
count.ordinal
collapse
Ωω{3}
ω-th 2-strong admissible = ω-th S3-admissible
ψ(Ωω{3}) - PTO of П14-CA0
46 symbol
count.ordinal
collapse
Ωω{4}
ω-th 2-strong admissible = ω-th S4-admissible
ψ(Ωω{4}) - PTO of П15-CA0
47 symbol
count.ordinal
cardinal equiv.
collapse
Ω{ω} = β0
first 2nd-order gap in the constructible universe = ramified
ω-Erdos
ψ(Ω{ω}) - PTO of Z2 and ZFC without powerset axiom and PTO of П20-CA0
48 symbol
count.ordinal
Ω{ω+1}
second 2nd-order gap in the constructible universe
49 symbol
count.ordinal

cardinal equiv.
Ω{ω×2} = β1
first 2nd-order gap of length 2 in the constructible universe
ω×2-Erdos
50 symbol
count.ordinal

cardinal equiv.
Ω{ω×3} = β3
first 2nd-order gap of length 3 in the constructible universe
ω×3-Erdos
51 symbol
count.ordinal
cardinal equiv.
Ω{Ω} = β+
first 2nd-order gap of length 1st admissible in the constructible universe
ω1-Erdos
52 symbol
count.ordinal
Ω{Ω{ω}}
first 2nd-order gap of length first 2nd-order gap in the constructible universe
47 symbol
count.ordinal
Ω{Ω{Ω{ω}}}
first 2nd-order gap of length first 2nd-order gap of length first 2nd-order gap in the constructible universe
53 symbol
count.ordinal
Ω{α} = Ω{Ω{Ω{Ω{Ω{...}}}}}
first 2nd-order gap of α-length in the constructible universe
54 symbol
count.ordinal
Ω{α}→I
first 2nd-order gap of inaccessible limit of α-length in the constructible universe
55 symbol
count.ordinal
Ω{α}→M
first 2nd-order gap of 1st Mahlo in which α-length in the constructible universe
56 symbol
count.ordinal

cardinal equiv.
Ω{α}→K
first 2nd-order gap of 2-admissible with П3-reflection over α-length in the constructible universe
Ramsey
57 symbol
count.ordinal

cardinal equiv.
Ω{α}→ K[K[K[K[K[...]]]]]
first 2nd-order gap of α-admissible with ω-le stability over α-length in the constructible universe
Jónsson
58 symbol
count.ordinal
Ω{α}→ K[α]→K[α]→K[α]→K[α]→...
first 2nd-order gap of 1st fixed point-α-admissible with α-le stability over α-length in the constructible universe
59 symbol
count.ordinal
Ω{α}→ Ω{1}
first 2nd-order gap of 1st strong admissible with nonprojectiblity over α-length in the constructible universe
60 symbol
count.ordinal
Ω{α}→Ω{α}
first 2nd-order gap of first 2nd-order gap of α-length in the constructible universe property over α-length in the constructible universe
61 symbol
count.ordinal
Ω{α}→Ω{α}→Ω{α}→Ω{α}...
first 2nd-order gap of 1st fixed point-α-length in the constructible universe
62 symbol
count.ordinal
cardinal equiv.
collapse
Ω{{1}}
admissible which is not locally countable
2-ineffable Ramsey
ψ(εΩ{{1}}+1) - PTO of KP+ω1
63 symbol
count.ordinal
Ω2{{1}}
2nd admissible which is not locally countable
64 symbol
count.ordinal
K{{1}} = K{{1}}[1]
2-admissible which is not locally countable
65 symbol
count.ordinal
K{{1}}[2]
3-admissible which is not locally countable
66 symbol
count.ordinal
Ω{{1}}{1}
1st strong admissible which is not locally countable = 1st S1-admissible which is not locally countable
67 symbol
count.ordinal
Ω2{{1}}{1}
2nd strong admissible which is not locally countable = 2nd S1-admissible which is not locally countable
68 symbol
count.ordinal
K{{1}}{1} = K{{1}}{1}[1]
strong 2-admissible which is not locally countable = 2-S1-admissible which is not locally countable
69 symbol
count.ordinal
K{{1}}{1}[2]
strong 3-admissible which is not locally countable = 3-S1-admissible which is not locally countable
70 symbol
count.ordinal
Ω{{1}}{2}
1st 2-strong admissible which is not locally countable = 1st S2-admissible which is not locally countable
71 symbol
count.ordinal
Ω{{1}}{3}
1st 3-strong admissible which is not locally countable = 1st S3-admissible which is not locally countable
72 symbol
count.ordinal
collapse
Ω{{1}}{ω}
first 2nd-order gap in the constructible universe on admissible which is not locally countable
ψ(Ω{{1}}{ω}) - PTO of П21-CA0
73 symbol
count.ordinal
Ω{{1}}{Ω{{1}}}
admissible which is not locally countable on admissible which is not locally countable
74 symbol
count.ordinal
collapse
Ω{{1}}{α}
admissible which is not locally countable on ... fixed point ... on admissible which is not locally countable
ψ(Ω{{1}}{α}) - PTO of П21-TR0
75 symbol
count.ordinal

cardinal equiv.
Ω{{2}}
admissibl