My question is, can a game contain itself as an option? and can it be a surreal number? For example $A=\{A|\}$ or $B=\{C|B\}$ where $C$ is a surreal number. from the point of view of games, it is equivalent to having the player play by doing nothing an effectively passing his turn to the second player. But I am not sure if this is okay from the point of view as surreal numbers. But as a number it is a problem, since for example to proof that $A=\{A|\}$ is a number, we will need to prove that all of its options are number, which means that we need to prove that $A$ is a number! So is it even possible to have a game with itself as an option? or will I have a "set which contains itself" paradox or something like that. Thanks!
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4$\begingroup$ Short answer: no. $\endgroup$– WojowuCommented Aug 9, 2017 at 16:55
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9$\begingroup$ Slightly longer answer: games that allow this are called "loopy" and are a different theory. Unless expressedly qualified as "loopy", games in the Conway sense are well-founded ("enders" or whatever Conway calls this) so they cannot be options of themselves (they satisfy the equivalent of the Foundation axiom in set theory). $\endgroup$– Gro-TsenCommented Aug 9, 2017 at 17:36
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$\begingroup$ Game $\{\;|\;\}$ is not the empty set. So $\{\;|\;\} \ne\{ \{\;|\;\}\;|\;\}$ $\endgroup$– Gerald EdgarCommented Aug 9, 2017 at 18:03
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2$\begingroup$ See Hypergame ... $\endgroup$– Nik WeaverCommented Aug 9, 2017 at 19:33
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A surreal number cannot have itself as an option, because numbers are defined inductively: if $L$ and $R$ are sets of numbers, then $\{L|R\}$ is a number. Notice that $L$ and $R$ have to already contain things that are numbers before we can say that $\{L|R\}$ is a number.
I suppose there would nothing a priori wrong with allowing games containing themselves as options, but it would obviously limit one's ability to analyze them as generalizations of surreal numbers. This is probably why in ONAG Conway writes "...we adopt the convention that in no game is there an infinite sequence of positions each of which is an option of its predecessor." (p.72) So although games containing themselves as options might conceivably model something interesting, they are explicitly ruled out by definition.
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1$\begingroup$ I think the question is a little more subtle than this. The inductive construction says that no number (or game) can be defined to have itself as an option, but doesn't exclude the possibility that a number might turn out, after the fact, to be equal to one of its options, since equality is a defined relation. However, since it's a general fact that every number is strictly greater than its left options and strictly less than its right options, this is also impossible. $\endgroup$ Commented Jul 29, 2018 at 4:34
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