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.



A game? is called determined iff some player has a winning 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 XX). The winning condition is then given as a subset AA of X ωX^\omega (for XωX\le\omega) and player 1 is considered to win iff the path chosen during the game is an element of AA.

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


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.


Last revised on July 25, 2023 at 17:10:18. See the history of this page for a list of all contributions to it.