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.