User blog:Jacquesbailhache/Simmons Ordinal Notation

Harold Simmons defined an ordinal notation which seems interesting to me, between Veblen and OCFs, whose limit is Bachmann Howard ordinal.

He uses the lambda calculus syntax, writing for example "f a b c" instead of "f(a,b,c)".

First, Simmons defines how to raise a function to an ordinal power : - g^0 z = z - g^(a+1) z = g(g^a z) - g^l z = V {g^a z | a < l } if l is a limit ordinal, where V denotes the pointwise supremum.

Then Simmons defines the following functions : Fix f z = f^w (z+1) Next = Fix (a -> w^a) [0] h = Fix (a -> h^a w) [1] h g = Fix (a -> h^a g w) [2] h g f = Fix (a -> h^a g f w) ... Δ[0] = w Δ[1] = Next w = ε_0 Δ[2] = [0] Next w = ζ_0 Δ[3] = [1] [0] Next w = Γ_0 Δ[4] = [2] [1] [0] Next w = Large Veblen Ordinal ... The limit of this sequence is Bachmann Howard Ordinal.

Veblen functions can be expressed with this notation, for example : φ(d,c,b,a) = ([0]^b (([1] [0])^c (([1]^2 [0])^d Next)))^(1+a) w

Klammersymbols can also be represented with Simmons notation.

References :



http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/Fruitful.pdf



http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/ordinal-notations.html