Axiom of real determinacy

{{short description|Axiom of set theory}}

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In mathematics, the axiom of real determinacy (abbreviated as AD_R) is an axiom in set theory.{{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 }} It states the following:

{{math theorem|Consider infinite two-person games with perfect information. Then, every game of length ω where both players choose real numbers is determined, i.e., one of the two players has a 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 integers; AD_R is inconsistent with the axiom of choice. It also implies the existence of inner models with certain large cardinals.

AD_R is equivalent to AD plus the axiom of uniformization.