Just a few days ago my seemingly eternal and recurrent fascination for Conway's *combinatorial game theory* (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent [survey](http://arxiv.org/PS_cache/math/pdf/0410/0410026v2.pdf), and began reading. 


Some old thoughts came to the surface from the archives of my memory. Here they are: 


**the class $SURREAL$ contains the class $ON$, and ordinals are the spine of $V$, the "universe of sets". So, pushing the analogy, can I say that combinatorial games generalize sets, or conversely sets are (special) combinatorial games?** 


If the answer is yes, can I even go further, and develop some foundational theory which starts *from games*, not sets, and then define ordinary sets as those special games?


This question can be broken down into 3 sub-questions:


1. **does there exist a treatment of combinatorial games as a first order axiomatic theory, presented without the recourse to sets?**
2. **what kind of games are ordinary ZF sets? (perhaps "solitaire" games, where the opponent doesn't do anything, or perhaps perfectly symmetric games). In other words, assuming 1) above, which interpretations of ZF are available inside CGT?**
3. **could one reformulate some familiar constructions of classical set theory in the language of CGT?**


Any material, thoughts, refs, on 1) -3)?


PS In this daydreaming I saw a picture of an extended universe where there is a double-cone of sets, V and -V, as in SURREAL there are positive and negative ordinals....



Sets as Combinatorial Games
I think the answer to the first question (in bold) is, "of course." Just as, quoting On Numbers and Games, 


* If $L, R$ are any two sets of numbers, and no member of $L$ is $\geq$ than any member of $R$, then there is a number $\{L|R\}$. All numbers are constructed this way


we can say 


* If $L, R$ are any two sets of games, then there is a game $\{L|R\}$. All games are constructed this way.


A good picture of a game is a tree with no infinite branches (so that we can argue by recursion; this corresponds to the phrase "all games are constructed this way", and is analogous to the Foundation axiom of set theory), where each edge is colored either Lavender or Red. If all edges are colored Lavender, say, then we have an ordinary (hereditary) set, and on the class of such games, the relation "$x$ is a member of $y$" is synonymous with "$x$ is an ($L$) option of $y$". 


It seems to me that a theory of games is perforce a theory of hereditary sets with not one but two membership relations $\in\_L$, $\in\_R$. You could of course give a theory of such just the same way you give a theory of sets, and it no more requires a pre-existing class of sets than the theory ZF requires a pre-existing class of sets. (Maybe I'm not exactly sure what you mean by "without recourse to sets".) The theory has two binary predicates $\in\_L, \in\_R$ where each separate predicate satisfies some axioms; they could be the ZF axioms, or the axioms of any other membership-based set theory. In combinatorial game theory, these two predicates don't interact (at the level of the axioms), but it might be interesting to contemplate theories where they *do* interact. 


Put this way, one can reformulate any construction of set theory in terms of game theory, but I don't see what advantage there would be. My own preferred development of the theory might be along the lines of Algebraic Set Theory (see the book by Joyal and Moerdijk). 


