You can find the definition for lambda notation, @x and $x here.
Also, when I say that A in lambda notation = B in standard ordinal notation, I mean that Ψ(@A) in lambda notation = Ψ($B) in standard ordinal notation.
Up to ???[]
Up to Ψ_I(0)[]
λ0 = Ω, Ψ(x,1) (x < Ω) = Ω1+x, and Ψ(λ0(x),1) (x ≥ Ω) = Ωx.
| Lambda notation | Standard ordinal notation |
|---|---|
| λ0 | Ω |
| Ψ(1,1) | Ω2 |
| Ψ(2,1) | Ω3 |
| Ψ(ω,1) | Ωω |
| Ψ(λ0(λ0),1) | ΩΩ |
| Ψ(λ0(λ02),1) | ΩΩ2 |
| Ψ(λ0(Ψ(1,1)),1) | ΩΩ2 |
| Ψ(λ0(Ψ(ω,1)),1) | ΩΩω |
| Ψ(λ0(Ψ(λ0(λ0),1)),1) | ΩΩΩ |
| Ψ(λ0(Ψ(λ0(Ψ(λ0(λ0),1)),1)),1) | ΩΩΩΩ |
Up to I[]
Ψ(λ0(Ψ(λ0,1)x),1) = ΨI(x), Ψ(λ0(Ψ(λ0,1)x)+y,1) = about ΩΨI(x)+y ,Ψ(λ0(@Ψ(λ0,1)),1) = ΨI($I), and Ψ(λ0,1) = I.
| Lambda notation | Standard ordinal notation |
|---|---|
| Ψ(λ0(Ψ(λ0,1)),1) | ΨI(0) |
| Ψ(λ0(Ψ(λ0,1))+1,1) | ΩΨI(0)+1 |
| Ψ(λ0(Ψ(λ0,1)+λ0),1) | ΩΨI(0)+Ω |
| Ψ(λ0(Ψ(λ0,1)+Ψ(λ0(Ψ(λ0,1)),1)),1) | ΩΨI(0)2 |
| Ψ(λ0(Ψ(λ0,1)+Ψ(λ0(Ψ(λ0,1))+1,1)),1) | ΩΩΨI(0)+1 |
| Ψ(λ0(Ψ(λ0,1)2),1) | ΨI(1) |
| Ψ(λ0(Ψ(λ0,1)Ψ(λ0(Ψ(λ0,1)),1)),1) | ΨI(ΨI(0)) |
| Ψ(λ0(Ψ(λ0,1)2),1) | ΨI(I) |
| Ψ(λ0(Ψ(λ0,1)22),1) | ΨI(I2) |
| Ψ(λ0(Ψ(λ0,1)3),1) | ΨI(I2) |
| Ψ(λ0(Ψ(λ0,1)Ψ(λ0,1)),1) | ΨI(II) |
| Ψ(λ0(ΨΨ(λ0,1)(0)),1) | ΨI(εI+1) |
| Ψ(λ0,1) | I |
Up to I_2[]
We do something similar to the last 2 sections here. Ψ(λ0+x,1) (x < Ω) = Ω1+x and Ψ(λ0+λ0(x),1) (x ≥ Ω) = Ωx. Then, Ψ(λ0+λ0(Ψ(λ02,1)x),1) = ΨI2(x), Ψ(λ0+λ0(Ψ(λ02,1)x)+y,1) = about ΩΨI2(x)+y ,Ψ(λ0+λ0(@Ψ(λ02,1)),1) = ΨI($I2), and Ψ(λ02,1) = I2.
Up to I(1,0)[]
Up to I(2,0)[]
Up to Ψ_I(1,0,0)(0)[]
Up to I(1,0,0)[]
Comparisons get more complex here because of how λ0 works inside of function F(x) = Ψ(x,1).