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.