Axiom of determinacy: Difference between revisions
→References
RhubarbJayde (talk | contribs) (Created page with "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 th...") |
Tag: Manual revert |
||
| (4 intermediate revisions by 2 users not shown) | |||
Line 1:
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-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is
Note that the determinacy of every topological game whose payoff set is closed,
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 [[Mahlo cardinal|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 theory|inner model]] containing both all [[Ordinal|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 [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\).
Assuming AD, \(\aleph_1\) and \(\aleph_2\) are [[Measurable cardinal|measurable]], \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.<ref>T. Jech, "About the Axiom of Choice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref><sup>p. 369</sup>
| |||