nLab axiom of determinacy

This axiom in set theory has important role implications in descriptive set theory (since regularity properties of certain classes of subsets of Polish spaces are implied by corresponding statements of determinacy). There are well-established relationships between the variants of the axiom and reflection principles/large cardinal axioms.

Definition

Definition

A game? is called determined iff some player has a winnig strategy. Typically in set theory Gale-Stewart games are considered, i. e. two players pick natural numbers (which might be required to be at most some $X$). The winning condition is then given as a subset $A$ of $X^\omega$ (for $X\le\omega$) and player 1 is considered to win iff the path chosen during the game is an elment of $A$.

The set set providing the winning condition is also called determined in the case that the game is determined.

Statement

The axiom of determinacy (AD) states that every Gale-Stewart game is determined. In ZF this axiom contradicts the axiom of choice.

Restrictions and relations to reflection principles

Determinacy for Borel sets can be proven in ZFC, however it cannot be proven in Zermelo set theory (without the axiom of replacement which is equivalent to Levy-Montague reflection?).

The existence of infinitely many Woodin cardinal?s implies the axiom of projective determinacy (PD) stating that all projective sets are determined.

The existence of infinitely many Woodin cardinals is equiconsistent to AD.

Reference

• Wikipedia, Axiom of determinacy