TITLE: A question about J.H. Conway's SURREAL NUMBERS
QUESTION [9 upvotes]: 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.

REPLY [6 votes]: Why on Earth do you feel the need to coerce Conway's system into some form of Set Theory?
Please go and read the Appendix to Part Zero of On Numbers and Games, which is a manifesto for the Mathematician's Lib Movement.
Conway's games are already formulated in much the same way as set theories are, the only difference being that they have left- and right-handed members.
Why do you think that an official one-handed set theory is any more suitable as a foundation than Conway's system free-standing?
Anyway, if you have coerced it into your preferred set theory, it becomes useless for anybody else's.  For example, Mike Shulman recently solved the problem of eliminating the double negations from every level, as part of the presentation of Conway Numbers in Homotopy Type Theory (in the final chapter of the book).