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A Simmons notation for transfinite ordinals

  1. Jan 12, 2018 #1
    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 ?
     
    Last edited: Jan 12, 2018
  2. jcsd
  3. Jan 17, 2018 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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