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It is my understanding that numbers & games have as there left & right options sets. I think this means that numbers & games are themselves sets. Gaps on the other hand can have (proper) classes for options. I think this also means that gaps themselves are (proper) classes. I'm curious what might be beyond numbers, games & gaps; additionally what kind of (set / category / type) theory would we need to work in to accomplish going further.

In Going beyond the surreal numbers it is mentioned in a comment that

If you work in a framework which allows talking about proper classes (like NBG or MK theories), then you can extend it one more time. If you have a theory, you can extend it twice. And so on.

Additionally, the answer states

Your proposal is to continue filling cuts after all ordinal birthdays are completed.

All such cuts will have cofinality Ord on one side or the other, and so each such cut will take a proper class to represent it. So the first thing to say is that there will be certain set-theoretic foundational difficulties with undertaking the construction. For example, this is not straight-forwardly a ZFC construction, but you could proceed in GBC for a step or so. To proceed much further, you will need stronger second-order set-theoretic axioms, such as the axiom ETR of elementary transfinite recursion, which allows one to undertake recursions on proper class well-founded relations whose rank exceeds Ord.

There is another question (Is there a “more powerful” form of set theory that would enable this?) which asks

I was wondering about this. In Conway's "On Numbers and Games", he discusses the "surreal numbers", and in one point mentions that they are full of "gaps". That the surreal number line is riddled with gaps. Namely, what he mentions is that these gaps occur for "cuts" between proper classes of surreal numbers, whereas ordinary surreal numbers are cuts between sets.

He then goes and mentions how that we cannot collect these together, it would be an "illegal" (undefined?) object in conventional set theory. Which makes me wonder -- could there exist some greater, more powerful form of set theory that could enable this kind of "higher-order collection" to exist? And then we could talk about the properties of "all surreal number plus all gaps in a single continuum". Or is there a good, fundamental reason that this simply cannot be done? If so, what is it? And if it can be done, what kind of properties would this monster have, anyways?

The answer to that question states a similar sentiment to the previous answer

You can extend ZFC to allow classes and 2-classes (classes of classes), but this makes things complicated and very delicate. Instead a common cure for the problem is to assume there exists a set model of ZFC (e.g. if there is an inaccessible cardinal), and talk about the surreals of that model from an external fashion. Namely the collection of all cuts is a set in the universe, but not a definable collection within that model. Again, this is full of delicate points.

There is another related question (Ultrainfinitism, or a step beyond the transfinite) which brings to mind oneiric numbers.

I'm not sure if there are any issues with games & set theory. It definitely seems as though there may be issues with something like the oneiric numbers however.

In summary, what base theory would allow us to go further in the surreal construction? Additionally, how would we go about our construction & what types of objects would we run into beyond numbers, games & gaps?

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    $\begingroup$ Don't the answers you cite suffice? If you are looking for precise set theories, for instance Ackermann Set Theory, or ZF+ reflective cardinal could be candidates. The thing is, you wouldn't want the surreal numbers in models of the stronger theory to be very different from those in models of ZF, because this would reflect as a difference of set theories. $\endgroup$
    – nombre
    May 24 '20 at 15:49
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    $\begingroup$ The "surreal numbers" business really doesn't seem to add much to your question. It seems you're basically asking about levels of "classes" in set theories, but as nombre says, it's not clear to me what remains unanswered. $\endgroup$
    – Mark S.
    May 24 '20 at 19:50
  • $\begingroup$ @nombre I'm not versed enough in set theory to understand the nuances of ZFC + Some Axiom vs NBG vs GBC vs MK vs Ackermann Set Theory. The answers cited makes it seem to me that going further is indeed possible but is "complicated" & "very delicate". I'm hoping to get a more in-depth picture beyond "it can be done". Will most likely post a bounty on this question once it is eligible. $\endgroup$
    – user784623
    May 24 '20 at 20:33
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    $\begingroup$ I thin Mark S.'s comment is on point. If you add numbers, you will get "longer numbers" whose length is a longer ordinal, and you'll probably have anything you would want already in a smaller set of surreal numbers such as $\mathbf{No}(\omega_1)$. In any case, you should note that there are always gaps in a linear order, so having a form of saturation as in the case of "regular" surreal numbers is already pretty good in terms of filling the gaps. $\endgroup$
    – nombre
    May 24 '20 at 21:03
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    $\begingroup$ Yes exactly, but you can fill each gap in $\mathbf{No}(\omega_1)$ while remaining in $\mathbf{No}(\omega_1+1)$. I want to say that the same would happen if you filled each gap in $\mathbf{No}(\kappa)$ in a reasonable stronger set theory where $V_{\kappa}$ is a model of ZFC. So maybe it is already enough to look inside $\mathbf{No}$ or $\mathbf{No}(\omega_1)$. Anyways there won't be any exotic things that occur that whose property wouldn't just be a statement about cardinals in said set theory. $\endgroup$
    – nombre
    May 24 '20 at 21:17
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Disclaimer: I'm not a professional mathematician & the following is just my personal speculation.


What base theory would allow us to go further in the surreal construction?

In the appendix to Part Zero of ONAG Conway says:

Plainly the proper set theory in which to perform a formalisation would be one with two kinds of membership.

It seems to us, however, that mathematics has now reached the stage where formalisation within some particular axiomatic set theory is irrelevant

What is proposed is instead that we give ourselves the freedom to create arbitrary mathematical theories of the kinds, but prove a metatheorem which ensures once and for all that any such theory could be formalised in terms of any of the standard foundational theories.

our Mathematicians’ Lib movement can be expressed directly in terms of the predicate calculus

So it seems like there are 3 main options:

  • find a flavor of set theory "with two kinds of membership"
  • prove (or find a proof) of a metatheorem
  • use predicate calculus

I particulary like what PlanetMath has to say on the subject:

surreal numbers were intended to be a basis for mathematics, not something to be embedded in set theory

So we could just work with the surreals in their natural habitat & not worry about making them fit into any other framework.

how would we go about our construction

We might start with "filling cuts after all ordinal birthdays are completed". This makes me think about the gap $\infty$ & how it is beyond $\mathbb{R}$ but less than positive infinite surreals. Perhaps there would be a gap between $\mathbf{On}$ & the oneiric numbers, something like: $$\Omega=\{\mathbf{On}||\mathbf{On}|\mathbf{On}\}$$ Drawing parallels to $$\infty=\{\mathbb{R}||\mathbb{R}|\mathbb{R}\}$$

what types of objects would we run into beyond numbers, games & gaps?

If numbers/games correlate to sets & gaps to classes, perhaps we could start getting into conglomerates? Conglomerates were "created to deal with 'collections' of classes". However, I'm not sure what you'd call cuts in conglomerates. We could also potentially work with Grothendieck universes (mentioned in the Wikipedia page for conglomerates). Again, not sure what you'd call a cut in a universe. It doesn't seem to me that there is any limit to how far we can go, or any way of knowing in advance what we will find in the vast expanse of terra incognita.

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