Editing Axiom of real determinacy

{{short description|Axiom of set theory}}
{{More citations needed|date=March 2024}}
In [[mathematics]], the '''axiom of real determinacy''' (abbreviated as '''AD<sub>R</sub>''') is an [[axiom]] in [[set theory]].<ref>{{Cite journal |last=Ikegami |first=Daisuke |last2=de Kloet |first2=David |last3=Löwe |first3=Benedikt |date=2012-11-01 |title=The axiom of real Blackwell determinacy |url=https://doi.org/10.1007/s00153-012-0291-x |journal=Archive for Mathematical Logic |language=en |volume=51 |issue=7 |pages=671–685 |doi=10.1007/s00153-012-0291-x |issn=1432-0665|doi-access=free }}</ref> It states the following:

{{math theorem|Consider infinite two-person [[Determinacy#Games|game]]s with [[perfect information]]. Then, every game of length [[ordinal number|ω]] where both players choose [[real number]]s is determined, i.e., one of the two players has a [[Determinacy#Winning strategies|winning strategy]].|name=Axiom}}

The axiom of real determinacy is a stronger version of the [[axiom of determinacy]] (AD), which makes the same statement about games where both players choose [[integer]]s; AD<sub>R</sub> is [[Consistency|inconsistent]] with the [[axiom of choice]]. It also implies the existence of [[inner model]]s with certain [[large cardinal]]s.

AD<sub>R</sub> is equivalent to AD plus the [[axiom of uniformization]].

== See also ==
* [[AD+|AD<sup>+</sup>]]
* [[Axiom of projective determinacy]]
* [[Topological game]]

[[Category:Axioms of set theory]]
[[Category:Determinacy]]

== References ==
{{Reflist}}{{settheory-stub}}