In his book *On Numbers and Games* (pg. 38) John Horton Conway makes the following remark, "But the collection of all gaps is not even a Proper Class, being an illegal object in most set theories." Is it true that the collection of all gaps is an "illegal object" in most set theories, and in what set theories is this collection not an illegal object? Conway's system No seems to handle gaps quite nicely, even without an axiomatization of set theory. 



A question regarding a remark of John Horton Conway
The gaps in $No$ are discussed in some detail in the Postscript to my paper “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in New Essays on Peirce’s Mathematical Philosophy, edited by Matthew Moore, Open Court Press, 2010, pp. 235-282, which may be downloaded from <http://www.ohio.edu/people/ehrlich/>. There are $ 2^{\aleph\_{On}}$ such gaps. Of these, 
$2^{\aleph\_{On}}$ have character $(On,On^{\*}).$ 


These are the gaps that lie within the various Archimedean classes of $No$, i.e. the subclasses of $No$ consisting of all elements that are Archimedean equivalent to one another. There are $\aleph\_{On}$ additional gaps, each of which has character $(\alpha, On^{\* })$ or character $ (On,\alpha^{\*} )$, for some regular initial ordinal $\alpha$. Roughly speaking, these gaps lie between Archimedean classes. 


Whereas in NBG one can discuss an individual gap in $No$, the class of all such gaps is not a legal class of NBG. It is, however, a legal class of Ackermann’s set theory (with the axiom of foundation for sets), which is a conservative extension of both ZFC and NBG (which I take to include global choice). Good discussions of Ackermann’s theory may be found in:


Fraenkel, Abraham A., Bar-Hillel, Yehoshua, and Lévy, Azriel: 1973, Foundations of Set Theory (Second Revised Edition), North-Holland Publishing Company, Amsterdam.


Lévy, Azriel: 1976, “The Role of Classes in Set Theory,” in Sets and Classes, edited by G.H. Müller, North-Holland Publishing Company, Amsterdam, pp. 173-215.


Lévy, Azriel and Vaught, Robert: 1961, “Principles of Partial Reflection in the Set Ttheories of Zermelo and Ackermann,” Pacific Journal of Mathematics 11, pp. 1045-1062.


Lévy, Azriel: 1959, “On Ackermann’s Set Theory,” The Journal of Symbolic Logic 24, pp. 154-166.


Reinhardt, William: 1970, “Ackermann’s Set Theory Equals ZF,” Annals of Mathematical Logic 2, pp. 189-259.


By the way, if one wants to work with Conway’s original construction, where each surreal number is a proper class, one may do so in Ackermann’s theory without appealing to “Scott’s trick,” which was referred to in one of the earlier responses. In fact, as Conway himself notes, appealing to Scott’s trick destroys the symmetry of his construction. If one wants to work in NBG, I think it is better to employ one of the two alternative constructions of $No$ introduced by the present author. One is a generalization of the von Neumann ordinal construction and the other is a generalization of the Dedekind cut construction based on the cuts of Cuesta Dutari. The first is discussed in my paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45, and the other was introduced in my “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25, (1988), pp. 7-16 ( Errata, 25 (1988), p. 233). Both papers may be downloaded from the above-mentioned site.


