Axiom of determinacy

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.

The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies that all subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property. The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the same cardinality as the full set of reals).

Furthermore, AD implies the consistency of Zermelo–Fraenkel set theory (ZF). Hence, as a consequence of the incompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF. It also implies the negation of the generalized continuum hypothesis (GCH): since GCH implies the axiom of choice, it is incompatible with AD (see below).