# Axiom of real determinacy

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

**Axiom** — 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.

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.

## See also[edit]

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