# Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.

Part 1 is about foundations. Much of the research that I've seen on the surreal numbers typically treats the foundational issues by either working in NBG set theory (Ehrlich's work) or formulating everything in ZFC and tiptoeing around a formal treatment of proper classes (Conway's work). Also, some authors, again such as Conway, prefer to de-emphasize (though not completely ignore) the foundational issues that arise and assume that the reader will work in a set theory "that can handle it."

I have yet to see any work that focuses on using Grothendieck universes/strongly inaccessible cardinals as part of the growing arsenal of tools used to handle surreal numbers. I'm especially interested in this approach for its potential benefits in studying topology and analysis over the surreals, as well as making it easier to talk about this stuff in the sort of ordinary set-theoretic terms that mathematicians already know about. So,

Question 1: Does anyone have any good references that deal with using ZFC + the Axiom of Universes (ZFC+AU) in working with surreal numbers?

I'm particularly curious to know what foundational issues may arise in ZFC+AU that don't arise in other formal systems like NBG. I'm also very curious to know how well theorems from one foundation can be translated to theother, such as translating Ehrlich's results from NBG to ZFC+AU, and vice versa. General references on the interplay between NBG and ZFC+AU are also welcome; working with surreals requires me to learn a bit more logic!

Part 2 is about the fact that large cardinals can be fun to think about, in a kind of mystical set-theoretic way, and they carry over to the surreals by giving you new and exotic surreal numbers. I'm very interested to see if anyone's studied these "large surreals," some of which I suspect will have very interesting properties. I'm pretty sure that however interesting you think your favorite large cardinal is, the set of surreals generated by it on that birthday has to be at least twice as interesting, and there's a whole zoo of large cardinals to look at. Measurable surreals sound particularly interesting to me. So,

Question 2: Does anyone have any good references that deal with the interaction of various large cardinal axioms on the field of surreal numbers, as well as the interesting properties some of these "large surreals" might have?

Many thanks, and I'd much appreciate any references to useful literature on this topic! I've read Conway's ONAG, Knuth's book, and many of Ehrlich's papers, and I'd like some guidance on what references to turn to next, particularly with respect to the topics listed here.

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I'm not aware of references that use universes in the study of surreal numbers. The reason --- for the non-existence of such references or for my non-awareness of them if they do exist --- is that there seems to be very little new to be said here. If $U$ is a Grothendieck universe, then there is a model of NBG (and in fact of the stronger Morse-Kelley theory) that has the elements of $U$ as its sets and the subsets of $U$ as its classes. Thus, whatever has been done in NBG (or MK) can be automatically translated to the setting of a Grothendieck universe.
Thanks. A question though, about "translating" things from NBG to ZFC+AU: consider models of ZFC with ill-founded $\omega$. The model never "knows" that $\omega$ is ill-founded, but the ambient theory does. Thus, the theorem "$\omega$ is well-founded" doesn't "translate" correctly from the perspective of the model of ZFC to the perspective of the ambient theory in which the model is constructed. (cont'd) –  Mike Battaglia Mar 24 at 9:11
This is what I'm worried about when "translating" results about NBG in ZFC+AU, because I don't want to continue using the perspective of NBG. Instead, I just want to create a universe $U$ large enough to contain all of the surreals that I care about, and then I want to prove theorems about $U$ in terms of the language of ZFC+AU instead. So how do I ensure that, for instance, Ehrlich's isomorphism between the "maximal" $\mathbb{R^\ast}$ and $\mathbf{No}$ in NBG holds for the universe in ZFC+AU itself, not just from the interpreted perspective of $U$ as a model of NBG? –  Mike Battaglia Mar 24 at 9:37
@Mike: I don't really understand your first comment, since it seems to vacillate between two translations, one from "the ambient theory" to ZFC+AU and one from NBG to AU. Of course, they're related, via a translation from the ambient theory to NBG. But that leaves the well-foundedness issue unclear for me. If the $\omega$ of a model of ZFC is ill-founded from the ambient theory's viewpoint, then so is the $\omega$ of the interpreted model of NBG, because it's exactly the same $\omega$. I see no problem translating between ZFC+AU and NBG, regardless of the ambient world. –  Andreas Blass Mar 24 at 21:45
@Mike: As with any interpretation, if X is any statement proved in the interpreted theory (in this case NBG), then the translation of X into the interpreting theory (in this case ZFU --- where the translation just replaces "set" and "class" with "element of $U$" and "subset of $U$", respectively) is a theorem of the interpreting theory. In particular, if $X$ is Ehrlich's theorem, expressed in the language of NBG, then the translation into set-language is provable in ZFU. –  Andreas Blass Mar 24 at 21:49