Simmons notation for transfinite ordinals

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jacquesb
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Joined: Fri Jan 12, 2018 10:54 pm UTC

Simmons notation for transfinite ordinals

Postby jacquesb » Sat Jan 13, 2018 2:36 pm UTC

Harold Simmons defined a simple but powerful notation for transfinite ordinals.

In summary :
- It uses lambda calculus formalism
- Fix f z = f^ω (z+1) = limit of z+1, f(z+1), f(f(z+1)), ... ; ; it is the least fixed point of f strictly greater than z
- Next = Fix (a -> ω^a) ; Next a is the least ε_b strictly greater than a, for example ε_0 = Next 0 = Next ω, and ε_a = Next^(1+a) 0 = Next^(1+a) ω
- [0] h = Fix (a -> h^a 0)
- [1] h g = Fix (a -> h^a g 0)
- [2] h g f = Fix (a -> h^a g f 0)
- ...

There is a correspondence with Veblen's φ function, for example φ(1+a,b) = ([0]^a Next)^(1+b) 0.

Simmons defines a sequence whose limit is the Bachmann-Howard ordinal :
- Δ[0] = ω
- Δ[1] = Next ω = ε_0 = φ(1,0)
- Δ[2] = [0] Next ω = ζ_0 = φ(2,0)
- Δ[3] = [1] [0] Next ω = Γ_0 = φ(1,0,0)
- Δ[4] = [2] [1] [0] Next ω = large Veblen ordinal
- ...

At first sight, it seems to me that ω could be replaced by 0 in these formulas.
For example :
- [0] Next 0 = Fix (a -> Next^a 0) 0 = limit of 1, Next 0 = ε_0, Next^ε_0 0 = ε_ε_0, ... = ζ_0
- [0] Next ω = Fix (a -> Next^a 0) ω = limit of ω+1, Next^(ω+1) 0 = ε_(ω+1), Next^(ε_(ω+1)) 0 = ε_ε_(ω+1), ... = ζ_0
Do you agree with this ?
Have you an idea about the reason for which Simmons chose to use ω instead of 0 in his formulas ?

Do you know if there are standard forms, fundamental sequences and a comparison algorithm for this notation ?
For the fundamental sequences I have an idea : perhaps we could take z+1, f(z+1), f(f(z+1)), ... as a fundamental sequence for Fix f z, is this correct?

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