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Consider games played with a set $S$ where the two players play elements of $S$ in turns for time $\omega$, "and then" the winner is chosen according to some pre-determined payoff set $P \subseteq S^\omega$. The axiom of determinacy says that such games of a certain type are determined (one of the players has a winning strategy). However, $ZF$ axioms are inconsistent with the statement that all such games are determined.

The wiki page for the axiom mentions a reason to believe in it, based on infinitary logic. I find the reason compelling, but I don't see it as restricted to a certain type of these games. Is there a way to have total determinacy? I mean a reasonable set of axioms of set theory that includes it, or perhaps some completely different system.

If not, how to convince myself of the essential wrongness of unrestricted determinacy? I know that over $ZF$ it implies the axiom of choice, and then we can construct an undetermined game with uncountable transfinite induction. Somehow this isn't enough for me to shake out the intial intuition. Also, it wouldn't work without the unobvious axiom of the powerset (but then using determinacy would be hard since only countable payoff sets could be constructed).

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  • $\begingroup$ Have you looked at the proof in ZF (without choice) that unrestricted determinacy is false? It uses a rather simple game: Player I selects a countable ordinal number $\xi$ on his first move. (His later moves are irrelevant.) Then player II can win by playing, in the remaining steps, a binary sequence coding a well-ordering of the natural numbers of order-type $\xi$. There's no winning strategy for I, and a winning strategy for II would give a one-to-one map of the countable ordinals to the reals, contradicting the (ordinary) axiom of determinacy. $\endgroup$
    – Andreas Blass
    Sep 22, 2021 at 16:34
  • $\begingroup$ @AndreasBlass a more detailed explanation (or a link to such) would be appreciated $\endgroup$
    – acupoftea
    Sep 22, 2021 at 17:04
  • $\begingroup$ I think this argument is in Mycielski's original paper on determinacy, "On the axiom of determinateness" Fund. Math. 53 (1963/64) (MathSciNet review MR0161787). $\endgroup$
    – Andreas Blass
    Sep 22, 2021 at 17:07

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To expand on Andreas Blass's comment:

The axiom of determinacy implies that there is no injection from $\omega_1$ to the reals. This follows from the facts:

a) AD implies that every uncountable set of reals contains a perfect subset.
b) If there is an injection from $\omega_1$ into the reals, then there is an uncountable set of reals without a perfect subset.

In the game described in the comments by Andreas, a winning strategy for Player II enables Player II to, for any countable ordinal $\alpha$ that Player I plays at the beginning, respond with a real number $r_\alpha$ coding a relation on the natural numbers with ordertype $\alpha$. The map $r\mapsto r_\alpha$ a injection from the countable ordinals to the reals.

With respect to the reasoning of infinitary De Morgan's Law, Joel Hamkins recently posted about this on Twitter, where he gives an example (of somewhat Gödelian flavor) of a game which can be shown to be undetermined in ZF alone.

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