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My quesion is: What set theory are the mathematicians who are developing the theory of these numbers working in-or are they, in fact, working outside any of the standard set theories?. Each surreal number is a mapping of an ordinal number into the pair (+,-) so that the collection S of all these numbers is a proper class. Moreover S is a real closed (ordered) field containing sub-collections which are ordinally similar to the class of ordinal numbers and to the set of real numbers (in their usual order). Since S is densely ordered but not order-complete, there exists an order-complete ordered collection C (constructed from the Dedekind cuts of S), which contains a dense sub-collection that is ordinally similar to S. Now the elements of C are proper classes and if we are going to have theorems about sub-collections of C (such as closed intervals), then the underlying set theory (if any) must be one that allows some proper classes to be elements of collections.

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These sorts of issues also come up in category theory. I have seen this notion of a hyper class that people speak about when collections of classes are needed. Another thing that is done ( if you are not too squeamish about foundations) is to admit a few inaccessible cardinals. I think 2 or 3 inaccessible cardinals should work for this, but I will leave that to people more knowledgeable about such issues. –  Lunasaurus Rex Sep 29 '11 at 20:24
The numbers themselves are sets. While ZFC does not allow proper classes as objects, it certainly allows discussing them. Inner model theory is a lot about it, so is forcing to some extent. Lastly, the NBG set theory is a conservative extension of ZFC which allows classes as objects. –  Asaf Karagila Sep 29 '11 at 20:27
Conway discusses this matter in the Appendix to Part Zero of On Numbers and Games. Recommended reading. –  Timothy Chow Sep 30 '11 at 0:34
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3 Answers

As you said, each surreal number can be regarded as a set, but the collection of all of them is a proper class. The set-theoretic issues involved in "developing the theory of these numbers" are the same as those involved in developing the theory of sets. For most purposes, ZFC suffices, since particular proper classes can be handled as "virtual classes" (essentially, formulas with set parameters). If one really needs to quantify over proper classes (whether of sets or of surreal numbers), then a set-class theory like Morse-Kelley becomes appropriate. If even higher types are needed, then I would be inclined to drop this one-step-at-a-time approach and instead assume that there is an inaccessible cardinal $\kappa$ and that Conway was really working in the universe of sets of rank $<\kappa$.

Note that difficulties with proper classes had to be faced in the foundations of category theory, and several approaches have been developed, including Grothendieck's universes and Feferman's approach based on the reflection principle of ZFC. As far as I can see, these approaches can be adapted to deal with the analogous problems that arise in connection with surreal numbers.

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Philip Ehrlich at Ohio University has written extensively on Conway's surreal numbers, and somewhere in his work he has the details for formalizing the theory of surreal numbers in NBG set theory. This should qualify as a "standard" set theory despite its use of classes as it's a conservative extension of ZFC, as mentioned above. His forthcoming paper for the Bulletin of Symbolic logic gives his paper

Absolutely saturated models. Fund. Math. 133 (1989), no. 1, 39–46.

as a reference for this formalization.



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Garabed, I believe the class of surreals can be encoded by a class formula in ZFC.

Surreal numbers are particular kinds of Conway games, and each Conway game can be expressed as a well-founded rooted tree of bounded (i.e., non-class) size equipped with a labeling of the edges by symbols $L$ and $R$. (The nodes are positions in the game, with the root at initial position, and an edge from a node to a child is labeled $L$ or $R$ according to whether the child is a left or right option of the node.) So the full structure of a Conway game is fully specified by a set, and the class of Conway games can be given by a ZFC class formula. The relation $\lt$ on games, and the predicate that says a game is a number, are recursive and can be given by formulas in ZFC. Similarly, the equality predicate on numbers is recursive and given by a ZFC formula.

I don't have On Numbers and Games immediately to hand, but my memory is that Conway discusses these issues.

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Thanks alot for your answers and suggestions. I think the main problem here is that the largest "large cardinal numbers" are elements of S because S contains a sub-collection ordinally similar to the class of all ordinal numbers. The theory of S has been much developed since Conway published "On Numbers and Games". Perhaps the theory of S can be formalized in Morse-Kelley set theory but only if that set theory allows the existence of continuously ordered collections containing dense sub-collections which are proper classes. –  Garabed Gulbenkian Oct 2 '11 at 17:46
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