# 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 $$ is said to be a winning strategy for I if it has the following property: if, after the play has gone ${n}_{0}{n}_{1}\mathrm{\dots}{n}_{M}$, I plays $\sigma ({n}_{0}\mathrm{\dots}{n}_{M})$ 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 $$, enumerate the uncountable closed subsets of the reals ${F}_{\alpha}$. Now construct two sequences $$ and $$ by choosing ${x}_{\alpha},{y}_{\alpha}$ as distinct points from ${F}_{\alpha}$ which are not in $$ (this is possible as each uncountable closed set has cardinality $\U0001d520$). 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 $\U0001d5ab[\mathbb{R}]$ 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 |