axiom of determinacy

When doing descriptive set theory, it is conventional to use either $\omega^{\omega}$ or $2^{\omega}$ as your space of “reals” (these spaces are homeomorphic to the irrationals and the Cantor set, respectively). Throughout this article, I will use the term “reals” to refer to $\omega^{\omega}$ .

Let $X\subseteq\omega^{\omega}$ be given and consider the following game on $X$ played between two players, I and II: I starts by saying a natural number; II hears this number and replies with another (or possibly the same one); I hears this and replies with another; etc. The sequence of numbers said (in the order they were said) is a point in $\omega^{\omega}$ . I wins if this point is in $X$ , otherwise II wins.

A map $\sigma:\omega^{<\omega}\to\omega$ is said to be a winning strategy for I if it has the following property: if, after the play has gone $n_{0}n_{1}\ldots n_{M}$ , I plays $\sigma(n_{0}\ldots n_{M})$ for each move, then I wins. A winning strategy for II is defined analogously.

The axiom of determinacy (AD) states that every such game is determined, that is either I or II has a winning strategy.

Using choice, a non-determined game can be constructed directly: for $\alpha<\mathfrak{c}$ , enumerate the uncountable closed subsets of the reals $F_{\alpha}$ . Now construct two sequences $\langle x_{\alpha}:\alpha<\mathfrak{c}\rangle$ and $\langle y_{\alpha}:\alpha<\mathfrak{c}\rangle$ by choosing $x_{\alpha},y_{\alpha}$ as distinct points from $F_{\alpha}$ which are not in $\{x_{\gamma},y_{\gamma}:\gamma<\alpha\}$ (this is possible as each uncountable closed set has cardinality $\mathfrak{c}$ ). Then the game on the set of all $x_{\alpha}$ s is non-determined.

From ZF+AD, one may prove many nice facts about the reals, such as: any subset is Lebesgue measurable, any subset has a perfect subset and the continuum hypothesis. ZF+AD also proves the axiom of countable choice.

AD itself is not taken seriously by many set theorists as a genuine alternative to choice. However, there is a weakening of AD (the axiom of quasi-projective determinacy, or QPD, which states that all games in $\mathsf{L}[\mathbb{R}]$ are determined) which is consistent with ZFC (in fact, it’s equiconsistent to a large cardinal axiom) which is a serious axiom candidate.

Title	axiom of determinacy
Canonical name	AxiomOfDeterminacy
Date of creation	2013-03-22 14:50:46
Last modified on	2013-03-22 14:50:46
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Axiom
Classification	msc 03E60
Classification	msc 03E15
Synonym	AD