# A Simmons notation for transfinite ordinals

1. Jan 12, 2018

### 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 ?

Last edited: Jan 12, 2018
2. Jan 17, 2018

### PF_Help_Bot

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