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 games of length ω with perfect information. AD states that every such game in which both players choose natural numbers 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.
Read more about Axiom Of Determinacy: Types of Game That Are Determined, The Axiom of Choice and The Axiom of Determinacy Are Incompatible, Infinite Logic and The Axiom of Determinacy, Large Cardinals and The Axiom of Determinacy
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