Axiom of real determinacy
# Axiom of real determinacy

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.

