axiom of determinacy

When doing descriptive set theory, it is conventional to use either ωω or 2ω as your space of “reals” (these spaces are homeomorphicMathworldPlanetmath to the irrationals and the Cantor setMathworldPlanetmath, respectively). Throughout this article, I will use the term “reals” to refer to ωω.

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

A map σ:ω<ωω is said to be a winning strategy for I if it has the following property: if, after the play has gone n0n1nM, I plays σ(n0nM) 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 α<𝔠, enumerate the uncountable closed subsets of the reals Fα. Now construct two sequences xα:α<𝔠 and yα:α<𝔠 by choosing xα,yα as distinct points from Fα which are not in {xγ,yγ:γ<α} (this is possible as each uncountable closed set has cardinality 𝔠). Then the game on the set of all xαs 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 hypothesisMathworldPlanetmath. 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 𝖫[] 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