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.
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. 235282, 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., BarHillel, Yehoshua, and Lévy, Azriel: 1973, Foundations of Set Theory (Second Revised Edition), NorthHolland Publishing Company, Amsterdam. Lévy, Azriel: 1976, “The Role of Classes in Set Theory,” in Sets and Classes, edited by G.H. Müller, NorthHolland Publishing Company, Amsterdam, pp. 173215. Lévy, Azriel and Vaught, Robert: 1961, “Principles of Partial Reflection in the Set Ttheories of Zermelo and Ackermann,” Pacific Journal of Mathematics 11, pp. 10451062. Lévy, Azriel: 1959, “On Ackermann’s Set Theory,” The Journal of Symbolic Logic 24, pp. 154166. Reinhardt, William: 1970, “Ackermann’s Set Theory Equals ZF,” Annals of Mathematical Logic 2, pp. 189259. 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. 145, and the other was introduced in my “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25, (1988), pp. 716 ( Errata, 25 (1988), p. 233). Both papers may be downloaded from the abovementioned site. 


The surreal numbers form a proper class (and even this is nontrivial, as each surreal number is defined as an equivalence class; however, Scott's trick can be used to present each surreal number by a set, the set of elements of the equivalence class of minimal rank). It follows that each gap in the surreal numbers is a pair of classes, and it is not clear to me whether these gaps can be coded by sets. Probably not in general. (You can identify each gap by its lower part, which gets rid of the pair of classes but still leaves you with a single class. Also, this is not a problem as there are ways to form a pair of two classes which is again a class.) The collection of all gaps is an object that consists of pairs of classes or just classes if you do it more cleverly. So it is a class of classes, if you wish. This is something that cannot be dealt with in ZFC. There are specific cases where it is possible to talk about a class of classes, but not in general. If I remember correctly, there was some discussion here on mathoverflow where it was pointed out that Harvey Friedman and other people have looked into set theories that can handle classes of classes and so on, but I have no reference for this right now. But how does No handle gaps? Note that even though No is defined by recursively filling something like Dedekind cuts, the construction is indexed by ordinals and at no stage of the construction you are dealing with all the surreal numbers. In each step you are handling things that look like gaps, but know only a set of surreal numbers. 


In NBG and similar set theories with classes, gaps are genuine objects of the theory. The collection of all gaps is not a class, but it is definable by a firstorder formula, so you can work with it in the theory in much the same way as one works with definable classes in ZFC. 

