FANDOM


EDIT: I abandoned this OCF (and this analysis) for a stronger one, that you can find here.

You can find the definition for my OCF here.

Also, when I say that an ordinal A equals another ordinal B, I mean that Ψ(@A) = Ψ(@B) and Ψ(@ΨA(x)) = Ψ(@ΨC(x)) (if it's not, then Ψ(@ΨA(x)) = Ψ(@ΨC+1(x))), where B = ΩC and @ is the rest of the ordinal inside of the function it's on (ends of parentheses (the '')''s) don't count as part of the ordinal inside of Ψ()).

If anyone understands standard ordinal notation with higher collapsed ordinals than weakly compact, please explain it to me so I can extend this analysis.

WIP!

Up to M

Up to Ψ_I(0)

This part is really easy. J(x) = Ωx.

My OCF Standard ordinal notation
J(0) Ω
J(1) Ω2
J(ω) Ωω
J(J(0)) ΩΩ
J(J(1)) ΩΩ2
J(J(J(0))) ΩΩΩ
J(J(J(J(0)))) ΩΩΩΩ
J(J(J(J(J(0))))) ΩΩΩΩΩ

Up to I

I's in ΨI(x) work similar to K(ω)'s in J(x). J(y+x) = ΩΨI(y)+x, J(K(ω)x) = ΨI(x), and J(@K(ω)) =  ΨI($I).

My OCF Standard ordinal notation
J(K(ω)) ΨI(0)
J(K(ω)+1) ΩΨI(0)+1
J(K(ω)+J(K(ω))) ΩΨI(0)2
J(K(ω)2) ΨI(1)
J(K(ω)J(K(ω))) ΨII(0))
J(K(ω)2) ΨI(I)
J(K(ω)22) ΨI(I2)
J(K(ω)3) ΨI(I2)
J(K(ω)K(ω)) ΨI(II)
J(K(ω)K(ω)K(ω)) ΨI(III)
J(ΨK(ω)(0)) ΨII+1)

Up to Ψ_I_2(0)

ΩI+x = K(ω+x) (for x < ω) and ΨKK(ω2)(x).

My OCF Standard ordinal notation
K(ω) I
K(ω+1) ΩI+1
K(ω+2) ΩI+2
ΨKK(ω2)(ω) ΩI+ω
ΨKK(ω2)(J(K(ω))) ΩI+ΨI(0)
ΨKK(ω2)(K(ω)) ΩI2
ΨKK(ω2)(K(ω+1)) ΩΩI+1
ΨKK(ω2)(ΨKK(ω2)(ω)) ΩΩI+ω
ΨKK(ω2)(ΨKK(ω2)(K(ω))) ΩΩI2
ΨKK(ω2)(ΨKK(ω2)(ΨKK(ω2)(K(ω)))) ΩΩΩI2

Up to Ψ_I(1,0)(0)

Ia works similar to ΨKK(ω2)(a), and ΩIa+bworks similar to ΨKΨKK(ω2)(a+1)(b).

Up to Ψ_I(2,0)(0)

Up to Ψ_I(1,0,0)(0)

Up to M

Up to Ξ[2,0]

Up to K

Up to U

Up to T

Beyond?

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