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(λ00),1) ΩΩ
Ψ(λ002),1) ΩΩ2
Ψ(λ0(Ψ(1,1)),1) ΩΩ2
Ψ(λ0(Ψ(ω,1)),1) ΩΩω
Ψ(λ0(Ψ(λ00),1)),1) ΩΩΩ
Ψ(λ0(Ψ(λ0(Ψ(λ00),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) ΨII(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(λ0Ψ(λ0,1)(0)),1) ΨII+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(λ00(x),1) (x ≥ Ω) = Ωx. Then, Ψ(λ00(Ψ(λ02,1)x),1) = ΨI2(x), Ψ(λ00(Ψ(λ02,1)x)+y,1) = about ΩΨI2(x)+y ,Ψ(λ00(@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).

Up to ???