Axiom of determinacy

The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the axiom of choice, since it implies that there is no well-ordering of the real numbers. However, its consistency strength relative to \(\mathrm{ZF}\) is very high.

Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice.

By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are strongly Mahlo, if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest inner model containing both all ordinals and all real numbers. Then the existence of both infinitely many Woodin cardinals and a measurable cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy.

Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of \(0^\sharp\), and boldface analytic determinacy is equiconsistent with the existence of \(r^\sharp\) for all real numbers \(r\).

Assuming AD, \(\aleph_1\) and \(\aleph_2\) are measurable, \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.^{p. 369}