Harold Simmons defined a simple but powerful notation for transfinite ordinals described in several articles available at(adsbygoogle = window.adsbygoogle || []).push({});

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

In summary :

- It uses lambda calculus formalism

- [itex] Fix f \zeta = f^\omega (\zeta+1) = [/itex] limit of [itex] \zeta+1, f (\zeta+1), f (f (\zeta+1)), ... [/itex] ; it is the least fixed point of f strictly greater than [itex] \zeta [/itex]

- [itex] Next = Fix (\alpha \mapsto \omega^\alpha) [/itex] ; [itex] Next\ \alpha [/itex] is the least [itex] \varepsilon_\beta [/itex] strictly greater than [itex] \alpha [/itex]. For example, [itex] \varepsilon_0 = Next\ 0 = Next\ \omega [/itex], and [itex] \varepsilon_\alpha = Next^{1+\alpha} 0 = Next^{1+\alpha} \omega [/itex].

- [itex] [0] h = Fix (\alpha \mapsto h^\alpha 0) [/itex]

- [itex] [1] h g = Fix (\alpha \mapsto h^\alpha g 0) [/itex]

- [itex] [2] h g f = Fix (\alpha \mapsto h^\alpha g f 0) [/itex]

- ...

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

Simmons defines a sequence whose limit is the Bachmann-Howard ordinal :

- [itex] \Delta[0] = \omega [/itex]

- [itex] \Delta[1] = Next\ \omega = \varepsilon_0 = \varphi(1,0) [/itex]

- [itex] \Delta[2] = [0] Next\ \omega = \zeta_0 = \varphi(2,0) [/itex]

- [itex] \Delta[3] = [1] [0] Next\ \omega = \Gamma_0 = \varphi(1,0,0) [/itex]

- [itex] \Delta[4] = [2] [1] [0] Next\ \omega = [/itex] large Veblen ordinal

- ...

At first sight, it seems to me that [itex] \omega [/itex] could be replaced by 0 in these formulas.

For example :

- [itex] [0] Next\ 0 = Fix (\alpha \mapsto Next^\alpha 0) 0 = [/itex] limit of [itex] 1, Next\ 0 = \varepsilon_0, Next^{\varepsilon_0} 0 = \varepsilon_{\varepsilon_0}, ... = \zeta_0 [/itex]

- [itex] [0] Next\ \omega = Fix (\alpha \mapsto Next^\alpha 0) \omega = [/itex] limit of [itex] \omega+1, Next^{\omega+1} 0 = \varepsilon_{\omega+1}, Next^{\varepsilon_{\omega+1}} 0 = \varepsilon_{\varepsilon_{\omega+1}}, ... = \zeta_0 [/itex]

Do you agree with this ?

Have you an idea about the reason for which Simmons chosed to use [itex] \omega [/itex] instead of 0 in his formulas ?

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# A Simmons notation for transfinite ordinals

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