Buchholz style Bachman style
ψ(I) = ψ(ψI(I)) = ψ(ψIIIII(...)))))) ψ(ψI(0)) = ψ(ΩΩΩΩ...)
ψ(ψI(I)+1) ψ(ψI(0)+1) = ψ(ψI(0))×ω
ψ(ψI(I)+Ω) = ψ(ψI(I)+ψ(ψI(I)+ψ(ψI(I)+...))) = εψ(ψI(I))+1 = ψ(ψI(I))ψ(ψI(I))ψ(ψI(I))... 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(ψI(I)+Ω2) = ψ(ψI(I)+ψ1I(I)+ψ1I(I)+...))) ψ(ψI(0)+Ω2) = ψ(ψI(0)+ψ1I(0)+ψ1I(0)+...)))
ψ(ψI(I)+Ωω) ψ(ψI(0)+Ωω)
ψ(ψI(I)+ΩΩ) ψ(ψI(0)+ΩΩ)
ψ(ψI(I)+ΩΩω) ψ(ψI(0)+ΩΩω)
ψ(ψI(I)+ΩΩΩω) ψ(ψI(0)+ΩΩΩω)
ψ(ψI(I)×2) = ψ(ψI(I)+ψI(I)) = ψ(ψI(I)+ΩΩΩΩ...) ψ(ψI(0)×2) = ψ(ψI(0)+ψI(0)) = ψ(ψI(0)+ΩΩΩΩ...)
ψ(ψI(I)×3) = ψ(ψI(I)+ψI(I)+ψI(I)) ψ(ψI(0)×3) = ψ(ψI(0)+ψI(0)+ψI(0))
ψ(ψI(I)×ω) ψ(ψI(0)×ω)
ψ(ψI(I)×Ω) = ψ(ψI(I)×ψ(ψI(I)×ψ(ψI(I)×....))) ψ(ψI(0)×Ω) = ψ(ψI(0)×ψ(ψI(0)×ψ(ψI(0)×....)))
ψ(ψI(I)2) ψ(ψI(0)2)
ψ(ψI(I)ω) ψ(ψI(0)ω)
ψ(ψI(I)Ω) = ψ(ψI(I)ψ(ψI(I))ψ(ψI(I))...) ψ(ψI(0)Ω) = ψ(ψI(0)ψ(ψI(0))ψ(ψI(0))...)
ψ(ψI(I)ψI(I)) ψ(ψI(0)ψI(0))
ψ(I+1) = ψ(ψI(I+1)) = ψ(ΩψI(I)+1) = ψ(εψI(I)+1) = ψ(ψI(I)ψI(I)ψI(I)...) ψ(ψI(0)ψI(0)ψI(0)...) = ψ(εψI(0)+1) = ψ(ΩψI(0)+1) = ψ(ΩΩΩΩ...+1)
ψ(ψI(I+1)+Ωω) = ψ(ΩψI(I)+1ω) ψ(ΩψI(0)+1ω)
ψ(ψI(I+1)+ψI(I)) = ψ(ΩψI(I)+1I(I)) = ψ(ΩψI(I)+1ΩΩΩ...) ψ(ΩψI(0)+1I(0)) = ψ(ΩψI(0)+1ΩΩΩ...)
ψ(ψI(I+1)+ψψI(I+1)I(I+1))) = ψ(ΩψI(I)+1ΩψI(I)+1ψI(I)+1)) = ψ(ΩψI(0)+1I(I)ψI(I)ψI(I)...) ψ(ΩψI(0)+1ΩψI(0)+1ψI(0)+1)) = ψ(ΩψI(0)+1I(0)ψI(0)ψI(0)...)
ψ(ψI(I+1)×2) = ψ(ψI(I+1)+ψI(I+1)) = ψ(ψI(I+1)+ψψI(I+1)I(I+1)+ψψI(I+1)I(I+1)+...))) = ψ(ΩψI(I)+1ψI(I)+1) = ψ(ΩψI(I)+1ΩψI(I)+1ψI(I)+1+...)) ψ(ΩψI(0)+1×2) = ψ(ΩψI(0)+1ψI(0)+1) = ψ(ΩψI(0)+1ΩψI(0)+1ψI(0)+1ΩψI(0)+1(...)))
ψ(ψI(I+1)2) = ψ(ψI(I+1)×ψI(I+1)) = ψ(ψI(I+1)×ψψI(I+1)I(I+1)×ψψI(I+1)I(I+1)×...))) = ψ(ΩψI(I)+1×ΩψI(I)+1) = ψ(ΩψI(I)+1×ψΩψI(I)+1ψI(I)+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(ψI(I+1)ψI(I+1)) = ψ(ψI(I+1)ψψI(I+1)I(I+1)ψψI(I+1)(...))) = ψ(ΩψI(I)+1ΩψI(I)+1) = ψ(ΩψI(0)+1ψΩψI(0)+1ψI(0)+1ψΩψI(0)+1(...))) ψ(ΩψI(0)+1ΩψI(0)+1) = ψ(ΩψI(0)+1ψΩψI(0)+1ψI(0)+1ψΩψI(0)+1(...)))
ψ(I+2) = ψ(ψI(I+2)) = ψ(ψI(I+1)ψI(I+1)...) = ψ(εψI(I+1)) = ψ(ΩψI(I)+2) = ψ(ΩψI(I)+1ΩψI(I)+1...) = ψ(εΩψI(I)+1) ψ(ΩψI(0)+2) = ψ(ΩψI(0)+1ΩψI(0)+1...) = ψ(εΩψI(0)+1)
ψ(I+ω) = ψ(ψI(I+ω)) = ψ(ΩψI(I)+ω) ψ(ΩψI(0)+ω)
ψ(I+Ω) = ψ(I+ψ(ψI(I+ψ(ψI(I+...))))) = ψ(ΩψI(I)+Ω) = ψ(ΩψI(I)+ψ(ΩψI(I)+...)) ψ(ΩψI(0)+Ω) = ψ(ΩψI(0)+ψ(ΩψI(0)+...))
ψ(I+Ωω) = ψ(I+ψωI(I+ψωI(I+...))))) = ψ(ΩψI(I)+Ωω) = ψ(ΩψI(I)+ψωψI(I)+...)) ψ(ΩψI(0)+Ωω)
ψ(I+ψI(I)) = ψ(ΩψI(I)+ψI(I)) = ψ(ΩψI(I)×2) ψ(ΩψI(0)×2)
ψ(I+ψI(I)+1) = ψ(ΩψI(I)×2+1) ψ(ΩψI(0)×2+1) = ψ(ΩψI(0)×2ΩψI(0)×2...) = ψ(εΩψI(0)×2)
ψ(I+ψI(I)+ψI(I)) = ψ(ΩψI(I)×3) ψ(ΩψI(0)×3)
ψ(I+ψI(I)×ω) = ψ(ΩψI(I)×ω) ψ(ΩψI(0)×ω)
ψ(I+ψI(I)×Ω) = ψ(I×ψ(ψI(I×ψ(ψI(I×...))))) = ψ(ΩψI(I)×Ω) = ψ(ΩψI(I)×ψ(ΩψI(I)×...)) ψ(ΩψI(0)×Ω) = ψ(ΩψI(0)×ψ(ΩψI(0)×...))
ψ(I+ψI(I)2) = ψ(ΩψI(I)2) ψ(ΩψI(0)2)
ψ(I+ψI(I)ω) = ψ(ΩψI(I)ω) ψ(ΩψI(0)ω)
ψ(I+ψI(I)Ω) = ψ(I+ψI(I)ψ(ψI(I))ψ(ψI(I))...) = ψ(ΩψI(I)Ω) = ψ(ΩψI(I)ψ(ΩψI(I)...)) ψ(ΩψI(0)Ω) = ψ(ΩψI(0)ψ(ΩψI(0)...))
ψ(I+ψI(I)ψI(I)) = ψ(ΩψI(I)ψI(I)) ψ(ΩψI(0)ψI(0))
ψ(I+ψψI(I+1)I(I+1))) = ψ(I+ψI(I)ψI(I)ψI(I)...) = ψ(ΩεψI(I)+1) = ψ(ΩψI(I)ψI(I)ψI(I)...) ψ(ΩψΩψI(0)+1ψI(0)+1)) = ψ(ΩεψI(0)+1) = ψ(ΩψI(0)ψI(0)ψI(0)...)
ψ(I+ψψI(I+1)I(I+2))) = ψ(ΩψΩψI(I)+1ψI(I)+2)) ψ(ΩψΩψI(0)+1ψI(0)+2))
ψ(I+ψψI(I+1)I(I+ψI(I)))) = ψ(ΩψΩψI(I)+1ψI(I)×2)) ψ(ΩψΩψI(0)+1ψI(0)×2))
ψ(I+ψψI(I+1)I(I+ψI(I)2))) = ψ(ΩψΩψI(I)+1ψI(I)2)) ψ(ΩψΩψI(0)+1ψI(0)2))
ψ(I+ψψI(I+1)I(I+ψI(I)ψI(I)))) = ψ(ΩψΩψI(I)+1ψI(I)ψI(I))) ψ(ΩψΩψI(0)+1ψI(0)ψI(0)))
ψ(I+ψψI(I+1)I(I+ψψI(I+1)I(I+1))))) = ψ(I+ψψI(I+1)I(I+ψI(I)ψI(I)ψI(I)...))) = ψ(ΩψΩψI(I)+1(ΩψΩψI(I)+1ψI(I)+1))) = ψ(ΩψΩψI(I)+1εψI(I)+1)) = ψ(ΩψΩψI(I)+1ψI(I)ψI(I)ψI(I)...)) 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(I+ψI(I+1)) = ψ(I+ψψI(I+1)I(I+ψψI(I+1)I(I+ψψI(I+1)I(I+...))))))) = ψ(ΩΩψI(I)+1) = ψ(ΩΩΩΩ...+1) = ψ(ΩψΩψI(I)+1ψΩψI(I)+1ψΩψI(I)+1(...)))) ψ(ΩΩψI(0)+1) = ψ(ΩΩΩΩ...+1) = ψ(ΩψΩψI(0)+1ψΩψI(0)+1ψΩψI(0)+1(...))))
ψ(I+ψI(I+1)+1) = ψ(ΩΩΩΩ...+1+1) = ψ(ΩΩψI(I)+1+1) ψ(ΩΩψI(0)+1+1) = ψ(ΩΩΩΩ...+1+1)
ψ(I+ψI(I+1)+ψψI(I+1)I(I+1)+ψI(I+1))) =  ψ(ΩΩψI(I)+1ΩψI(I)+1ΩψI(I)+1)) ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1))
ψ(I+ψI(I+1)×2) = ψ(I+ψI(I+1)+ψψI(I+1)I(I+1)+ψI(I+ψψI(I+1)I(I+ψI(I+1)+...))))) = ψ(ΩΩψI(I)+1×2) = ψ(ΩΩψI(I)+1ψI(I)+1) = ψ(ΩΩψI(I)+1ΩψI(I)+1ΩψI(I)+1+...)) ψ(ΩΩψI(0)+1×2) = ψ(ΩΩψI(0)+1ψI(0)+1) = ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1+...))
ψ(I+ψψI(I+2)I(I+2))) = ψ(I+ψI(I+1)ψI(I+1)ψI(I+1)...) = ψ(ΩψΩψI(I)+2ψI(I)+2)) = ψ(ΩΩψI(I)+1ΩψI(I)+1ΩψI(I)+1...) ψ(ΩψΩψI(0)+2ψI(0)+2)) = ψ(ΩΩψI(0)+1ΩψI(0)+1ΩψI(0)+1...)
ψ(I+ψI(I+2)) = ψ(I+ψψI(I+2)I(I+ψψI(I+2)I(I+ψψI(I+2)I(I+...))))))) = ψ(ΩΩψI(I)+2) = ψ(ΩψΩψI(I)+2ψΩψI(I)+2ψΩψI(I)+2(...)))) ψ(ΩΩψI(0)+2) = ψ(ΩψΩψI(0)+2ψΩψI(0)+2ψΩψI(0)+2(...))))
ψ(I+ψI(I+ω)) = ψ(ΩΩψI(I)+ω) ψ(ΩΩψI(0)+ω)
ψ(I+ψI(I+Ω)) = ψ(I+ψI(I+ψ(I+ψI(I+ψ(I+ψI(I+...)))))) = ψ(ΩΩψI(I)+Ω) = ψ(ΩΩψI(I)+ψ(ΩΩψI(I)+...)) ψ(ΩΩψI(0)+Ω) = ψ(ΩΩψI(0)+ψ(ΩΩψI(0)+...))
ψ(I+ψI(I+Ωω)) = ψ(I+ψI(I+ψω(I+ψI(I+ψω(I+ψI(I+...)))))) = ψ(ΩΩψI(I)+Ωω) = ψ(ΩΩψI(I)+ψωΩψI(I)+...)) ψ(ΩΩψI(0)+Ωω)
ψ(I+ψI(I+ψI(I))) = ψ(ΩΩψI(I)×2) = ψ(ΩΩψI(I)+ψI(I)) ψ(ΩΩψI(0)×2) = ψ(ΩΩψI(0)+ψI(0))
ψ(I+ψI(I+ψI(I+1))) = ψ(ΩΩΩψI(I)+1) ψ(ΩΩΩψI(0)+1)
ψ(I+ψI(I+ψI(I+ψI(I)))) = ψ(ΩΩΩψI(I)×2) ψ(ΩΩΩψI(0)×2)
ψ(I+ψI(I+ψI(I+ψI(I+1)))) = ψ(ΩΩΩΩψI(I)+1) ψ(ΩΩΩΩψI(0)+1)
ψ(I×2) = ψ(I+I) = ψ(I+ψI(I+ψI(I+ψI(I+ψI(I+...))))) = ψ(ΩΩΩ...ψI(I)+1) ψ(ψI(1)) = ψ(ΩΩΩ...ψI(0)+1)
ψ(I×2+1) = ψ(ΩψI(I×2)+1) ψ(ΩψI(1)+1)
ψ(I×2+ψI(I)) = ψ(ΩψI(I×2)+ψI(I)) ψ(ΩψI(1)+ψI(0))
ψ(I×2+ψI(I×2)) = ψ(ΩψI(I×2)×2) ψ(ΩψI(1)×2)
ψ(I×2+ψI(I×2+1)) = ψ(ΩΩψI(I×2)+1) ψ(ΩΩψI(1)+1)
ψ(I×2+ψI(I×2+ψI(I×2))) = ψ(ΩΩψI(I×2)×2) ψ(ΩΩψI(1)×2)
ψ(I×3) = ψ(I×2+I) = ψ(I×2+ψI(I×2+ψI(I×2+ψI(I×2+ψI(...))))) = ψ(ΩΩΩ...ψI(I×2)+1) ψ(ψI(2)) = ψ(ΩΩΩ...ψI(1)+1)
ψ(I×ω) ψ(ψI(ω))
ψ(I×Ω) = ψ(I×ψ(ψI(I×ψ(ψI(I×ψ(ψI(I×ψ(ψI(...))))))))) ψ(ψI(Ω)) = ψ(ψI(ψ(ψI(ψ(ψI(ψ(ψI(...))))))))
ψ(I×Ω2) = ψ(I×ψ1I(I×ψ1I(I×ψ1I(I×ψ1I(...))))))))) ψ(ψI2)) = ψ(ψI1I1I1I(...))))))))
ψ(I×Ω) = ψ(I×ψωI(I×ψωI(I×ψωI(I×ψωI(...))))))))) ψ(ψIω)) = ψ(ψIωIωIωI(...))))))))
ψ(I×ψI(I)) = ψ(I×ΩΩΩΩ...) ψ(ψII(0))) = ψ(ψIΩΩΩ...))
ψ(I×ψI(I)+1) = ψ(ΩψI(I×ψI(I))+1) ψ(ΩψII(0))+1)
ψ(I×ψI(I)+ψI(I×ψI(I))) = ψ(ΩψI(I×ψI(I))×2) ψ(ΩψII(0))×2)
ψ(I×(ψI(I)+1)) = ψ(I×ψI(I)+ψI(I×ψI(I)+ψI(I×ψI(I)+...))) = ψ(ΩΩΩ...ψI(I×ψI(I))+1) ψ(ψII(0)+1))
ψ(I×(ψI(I)+ψI(I))) ψ(ψII(0)×2))
ψ(I×ψI(I+1)) = ψ(I×(ψI(I)ψI(I)ψI(I)...)) ψ(ψIψI(0)+1))
ψ(I×ψI(I×2)) ψ(ψII(1)))
ψ(I×ψI(I×ψI(I))) ψ(ψIII(0))))
ψ(I2) = ψ(I×I) = ψ(I×ψI(I×ψI(I×ψI(I×ψI(I×...))))) ψ(I) = ψ(ψI(I)) = ψ(ψIIII(...)))))
ψ(ψI(I2)+1) = ψ(I2)×ω ψ(I+1) = ψ(I)×ω = ψ(ψIIII(...))))+1)
ψ(ψI(I2)+Ω) = ψ(ψI(I2)+ψ(ψI(I2)+ψ(ψI(I2)+ψ(ψI(I2)+...)))) = ψ(I2)ψ(I2)ψ(I2)... ψ(I+Ω) = ψ(ψI(I+ψ(ψI(I+ψ(ψI(I+ψ(ψI(I+...)))))))) = ψ(I)ψ(I)ψ(I)...
ψ(ψI(I2)+Ω2) = ψ(ψI(I2)+ψ2I(I2)+ψ2I(I2)+ψ2I(I2)+...)))) ψ(I+Ω2) = ψ(ψI(I+ψ2I(I+ψ2I(I+ψ2I(I+...))))))))
ψ(ψI(I2)+Ωω) = ψ(ψI(I2)+ψωI(I2)+ψωI(I2)+ψωI(I2)+...)))) ψ(I+Ωω) = ψ(ψI(I+ψωI(I+ψωI(I+ψωI(I+...))))))))
ψ(ψI(I2)+ψI(I)) = ψ(ψI(I2)+ΩΩΩΩ...) ψ(I+ψI(0)) = ψ(I+ΩΩΩΩ...)
ψ(ψI(I2)+ψI(I×2)) ψ(I+ψI(1))
ψ(ψI(I2)+ψI(I×ψI(I))) ψ(I+ψII(0)))
ψ(ψI(I2)×2) =ψ(ψI(I2)+ψI(I2)) = ψ(ψI(I2)+ψI(I×ψI(I×ψI(I×ψI(I×...))))) ψ(I+ψI(I)) = ψ(I+ψIIII(...)))))
ψ(ψI(I2)×3) ψ(I+ψI(I)×2)
ψ(ψI(I2)×ω) ψ(I+ψI(I)×ω)
ψ(I2+1) = ψ(ψI(I2+1)) = ψ(ψI(I2)ψI(I2)ψI(I2)...) = ψ(εψI(I2)+1) = ψ(ΩψI(I2)+1) ψ(I+ψI(I)ψI(I)ψI(I)...) = ψ(I+εψI(I)+1) = ψ(I+ΩψI(I)+1)
ψ(I2+2) = ψ(ΩψI(I2)+2) = ψ(ψI(I2+2)) ψ(I+ΩψI(I)+2)
ψ(I2I(I2)) = ψ(ΩψI(I2)×2) ψ(I+ΩψI(I)×2)
ψ(I2+I) = ψ(I2I(I2I(I2I(I2+...)))) ψ(I+ψI(I+1)) = ψ(I+ΩΩΩ...ψI(I)+1) = ψ(ψIIII(...)))+1))
ψ(I2+I×ψI(I2+I)) = ψ(I2+I+ψI(I2+I+ψI(I2+I+ψI(I2+I+...)))) ψ(I+ψI(I+ψI(I+1))) = ψ(I+ψI(I+ΩΩΩ...ψI(I)+1)) = ψ(ψIIII(...))+1)))
ψ(I2×2) = ψ(I2+I2) = ψ(I2+I×I) = ψ(I2+I×ψI(I2+I×ψI(I2+I×ψI(I2+I×...)))) ψ(I×2) = ψ(I+I) = ψ(I+ψI(I+ψI(I+ψI(I+...)))) = ψ(ψIIII(...ψI(I)+1...)))))
ψ(I2×ω) ψ(I×ω)
ψ(I2×Ω) = ψ(I2×ψ(ψI(I2×ψ(ψI(I2×...))))) ψ(I×Ω) = ψ(I×ψ(ψI(I×ψ(ψI(I×...)))))
ψ(I2×ψI(I)) = ψ(I2×ΩΩΩΩ...) ψ(I×ψI(0)) = ψ(I×ΩΩΩΩ...)
ψ(I2×ψI(I2)) ψ(I×ψI(I))
ψ(I3) = ψ(I2×I) = ψ(I2×ψI(I2×ψI(I2×ψI(I2×...)))) ψ(I2) = ψ(I×I) = ψ(I×ψI(I×ψI(I×ψI(I×...))))
ψ(I4) = ψ(I3×I) = ψ(I3×ψI(I3×ψI(I3×ψI(I3×...)))) ψ(I3) = ψ(I2×I) = ψ(I2×ψI(I2×ψI(I2×ψI(I2×...))))
ψ(Iω) - catching point ψ(Iω) - catching point