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jacquesb
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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
- [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 ?
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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