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.
[Math] A question about J.H. Conway’s SURREAL NUMBERS
set-theorysurreal-numbers
Best Answer
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.