Simmons notation for transfinite ordinals

[jacquesb]

Harold Simmons defined a simple but powerful notation for transfinite ordinals described in several articles available at
http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/ordinal-notations.html .

In summary :
- It uses lambda calculus formalism
- $Fix f \zeta = f^\omega (\zeta+1) =$ limit of $\zeta+1, f (\zeta+1), f (f (\zeta+1)), ...$ ; it is the least fixed point of f strictly greater than $\zeta$
- $Next = Fix (\alpha \mapsto \omega^\alpha)$ ; $Next\ \alpha$ is the least $\varepsilon_\beta$ strictly greater than $\alpha$. For example, $\varepsilon_0 = Next\ 0 = Next\ \omega$, and $\varepsilon_\alpha = Next^{1+\alpha} 0 = Next^{1+\alpha} \omega$.
- $[0] h = Fix (\alpha \mapsto h^\alpha 0)$
- $[1] h g = Fix (\alpha \mapsto h^\alpha g 0)$
- $[2] h g f = Fix (\alpha \mapsto h^\alpha g f 0)$
- ...

There is a correspondence with Veblen's $\varphi$ function, for example $\varphi(1+\alpha,\beta) = ([0]^\alpha Next)^{1+\beta} 0$.

Simmons defines a sequence whose limit is the Bachmann-Howard ordinal :
- $\Delta[0] = \omega$
- $\Delta[1] = Next\ \omega = \varepsilon_0 = \varphi(1,0)$
- $\Delta[2] = [0] Next\ \omega = \zeta_0 = \varphi(2,0)$
- $\Delta[3] = [1] [0] Next\ \omega = \Gamma_0 = \varphi(1,0,0)$
- $\Delta[4] = [2] [1] [0] Next\ \omega =$ large Veblen ordinal
- ...

At first sight, it seems to me that $\omega$ could be replaced by 0 in these formulas.
For example :
- $[0] Next\ 0 = Fix (\alpha \mapsto Next^\alpha 0) 0 =$ limit of $1, Next\ 0 = \varepsilon_0, Next^{\varepsilon_0} 0 = \varepsilon_{\varepsilon_0}, ... = \zeta_0$
- $[0] Next\ \omega = Fix (\alpha \mapsto Next^\alpha 0) \omega =$ limit of $\omega+1, Next^{\omega+1} 0 = \varepsilon_{\omega+1}, Next^{\varepsilon_{\omega+1}} 0 = \varepsilon_{\varepsilon_{\omega+1}}, ... = \zeta_0$
Do you agree with this ?
Have you an idea about the reason for which Simmons chosed to use $\omega$ instead of 0 in his formulas ?

 

[Valkarie]

The reason Simmons chose to use omega instead of 0 in his formulas is that omega is a special kind of ordinal. Omega is the smallest infinite ordinal and thus can be used as a starting point for defining larger ordinals. Using omega as the starting point is more convenient than using 0, as all the definitions involving omega will work correctly with the same formulas if omega is replaced by any larger ordinal. This is not necessarily true if 0 is used instead of omega.