Editing Axiom of determinacy (section)

{{Short description|Possible axiom for set theory}}
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 game]]s of length [[ω (ordinal number)|ω]]. AD states that every game of a [[Axiom of determinacy#Types of game that are determined|certain type]] is [[determined game|determined]]; that is, one of the two players has a [[winning strategy]].

Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model [[L(R)]] of a set theory, which accepts only a weak form of the [[axiom of choice]] (AC) but contains all [[real number|real]] and all [[ordinal number]]s. Some consequences of AD followed from theorems proved earlier by [[Stefan Banach]] and [[Stanisław Mazur]], and [[Morton Davis]]. [[Mycielski]] and [[Stanisław Świerczkowski]] contributed another one: AD implies that all sets of [[real number]]s are [[Lebesgue measurable]]. Later [[Donald A. Martin]] and others proved more important consequences, especially in [[descriptive set theory]]. In 1988, [[John R. Steel]] and [[W. Hugh Woodin]] concluded a long line of research. Assuming the existence of some [[Large cardinal|uncountable cardinal numbers analogous to ℵ<sub>0</sub>]], they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).