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I have not previously seen anything online that dives deeply into On:

In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. -Source

I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):

Epsilon-Nought $$\varepsilon_{0}=\{\omega,\omega^\omega,\omega^{\omega^\omega},...|\}$$ Feferman-Schutte Ordinal $$\Gamma_0=\phi_{1,0}(0)=\{\phi_0(0),\phi_{\phi_0(0)}(0),\phi_{\phi_{\phi_0(0)}(0)}(0),...|\}$$ Small Veblen Ordinal $$SVO=\{\phi_1(0), \phi_{1,0}(0), \phi_{1,0,0}(0),...|\}$$ Bachmann-Howard Ordinal $$BHO=\{\psi(\Omega),\psi(\Omega^\Omega),\psi(\Omega^{\Omega^\Omega}),...|\}$$

Additionally, any online resources related to On would be greatly appreciated.

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    $\begingroup$ In all those cases, the Conway's bracket notation could be replaced by the usual notion of supremum so I don't think there's much to gain here from that. $\endgroup$
    – nombre
    Commented Nov 21, 2018 at 11:38
  • $\begingroup$ @nombre I was hoping you might make an appearance! I am a) trying to confirm that, as written, everything (LVO excluded) is correct b) seek sources about On (surordinals?). I would like to eventually be able to perform calculations such as $\{\varepsilon_0 | \varepsilon_0\}$, $\{SVO|LVO\}$, $\{\Gamma_0|\Gamma_1\}$, etc. $\endgroup$
    – user820789
    Commented Nov 21, 2018 at 21:43
  • $\begingroup$ @nombre "the relationship between the two is deep enough for me to imagine that it could be wise at times to view ordinals as surreals to gain insight on ordinals" source $\endgroup$
    – user820789
    Commented Nov 29, 2018 at 4:23
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    $\begingroup$ I think it could be wise at times yes, and for instance I was able to anwser a question of mine about compositions of ordinals using inspiration from surreal numbers; but I wouldn't qualify this as a significant insight on ordinal numbers that naturally comes from surreal numbers. $\endgroup$
    – nombre
    Commented Nov 29, 2018 at 9:08

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I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $\Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.

Regarding your definitions, the function $\phi_{\gamma}(\alpha)$ should also be greater than every ordinal $\phi_{\eta}^{\circ n}(\phi_{\gamma}(\beta)+1)$ for $\eta < \gamma$, $n \in \mathbb{N}$ and $\beta<\alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.

Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $\phi_{\gamma}$ can be extended to $\mathbf{No}$ in a natural way.

For $x=\{L\ | \ R\} \in \mathbf{No}$, you must know about $\omega^x=\phi_0(x)=\{0,\mathbb{N}\ \phi_0(L)\ | \ 2^{-\mathbb{N}} \ \phi_0(R)\}$. Then the class of numbers $e$ such that $\omega^e=e$ is parametrized by $\varepsilon_x=\phi_1(x):=\{\phi_0^{\circ \mathbb{N}}(0),\phi_0^{\circ \mathbb{N}}(\phi_1(L)+1)\ | \ \phi_0^{\circ \mathbb{N}}(\phi_1(R)-1)\}$, and one can keep going on. At every stage $0<\gamma$, the function $\phi_{\gamma}$ parametrizes the class of numbers $e$ with $\forall \eta < \gamma,\phi_{\eta}(e)=e$.

As for sources on $\mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.


edit: to be more explicit, Conway's so-called $\omega$-map is defined inductively as $x \longmapsto \omega^x:=\{0,n \ \omega^{x_L}:n \in \mathbb{N} \wedge x' \in x_L \ | \ 2^{-n} \ \omega^{x''}:n \in \mathbb{N} \wedge x'' \in x_R\}$ where $x=\{x_L \ | \ x_R\}$. This is done so as to yield $r \omega^x < s \omega^y$ whenever $x<y$ and $r,s$ are strictly positive real numbers.

For $\phi_1$, this is $\phi_1(x):=\{\phi_0^{\circ n}(0),\phi_0^{\circ n}(\phi_1(x')+1): x' \in x_L \wedge n \in \mathbb{N} \ | \ \phi_0^{\circ n}(\phi_1(x'')-1): x' \in x_R \wedge n \in \mathbb{N}\}$, where $f^{\circ n}$ denotes the $n$-fold composition of a function $f$ with itself.

You can find both of those in Conway's On Numbers and Games, Chapter 3 and in Gonshor's An Introduction to the Theory of Surreal Numbers, Chapters 5 and 9. This is also discussed in some detail in Sections 5 and 6 of the pre-print Surreal Substructures (the formula for fixed points parametrizations is Remark 6.23).

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  • $\begingroup$ I still don't understand the parameterization in this answer. If you happen to have a reference I would great appreciate it. Wrt new insight on ordinal numbers - I suppose I just find it interesting that in surreals you can do something like $\frac{\sqrt{\omega}}{\varepsilon_0}$, which as far as I understand, isn't something you can do normally with ordinals. $\endgroup$
    – user820789
    Commented May 26, 2020 at 22:22
  • $\begingroup$ @meowzz I edited with a bit more information. $\endgroup$
    – nombre
    Commented May 27, 2020 at 15:31
  • $\begingroup$ Thanks for the additional info. I have a copy of ONAG & really like it (although I often have a hard time following Conway). I haven't managed to get Gonshor yet & substructures pre-print is way over my head. I tried to work out some basic examples of what you provided here & from ONAG, however I really don't understand it. I was wondering if you might be willing to work out an example case so I can try to follow along. If it's too much of hassle, no worries. Thanks again for this (& lots of other amazing answers I have seen by you). $\endgroup$
    – user820789
    Commented May 28, 2020 at 21:34
  • $\begingroup$ @meowzz You're welcome. Conway's writing is great but he tends not to go into details. He is particularly elusive regarding those fixed point functions. Gonshor however is rather the opposite so you should find it easier to follow. What exactly are you looking for an example of? $\endgroup$
    – nombre
    Commented May 29, 2020 at 8:50

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