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Has anyone studied Combinatorial game theory in non-standard models?

In particular, we can work in either non-standard models of set theory, or we can work in non-standard models of arithmetic, where $l$ and $r$ are binary predicates representing Left and Rights moves.

In particular, I think that studying Sylver coinage in non-standard models of PA would be interesting.

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    $\begingroup$ Can you clarify what you mean exactly by "non-standard models of set theory"? The term is not unambiguous. Also, try to give some indication why you think studing Sylver coinage this way would be interesting. $\endgroup$ – Mikhail Katz Feb 22 '18 at 10:25
  • $\begingroup$ Is there a particular approach to a nonstandard model of PA you have in mind? I don't see how you could get interesting Sylver Coinage results in, say, a countable model or an ultrapower model. $\endgroup$ – Mark S. Feb 22 '18 at 11:19
  • $\begingroup$ @MikhailKatz like, a model that doesn't equal $(V,\in)$ (according to Kelly Morse set theory or the like). $\endgroup$ – PyRulez Feb 22 '18 at 13:25
  • $\begingroup$ @MikhailKatz also, using a nonstandard model means you can analysis non standard positions. Basically, you can analyze positions involving infinity. $\endgroup$ – PyRulez Feb 22 '18 at 13:29
  • $\begingroup$ I am still not sure whether you are proposing the use of a nonstandard theory or nonstandard model. $\endgroup$ – Mikhail Katz Feb 22 '18 at 13:30

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