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, 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:
does there exist a treatment of combinatorial games as a first order axiomatic theory, presented without the recourse to sets?
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?
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....