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.

单词	AxiomOfDeterminacy
释义	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.
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