Axiom of real determinacy Information

From Wikipedia

In mathematics, the axiom of real determinacy (abbreviated as ADR) 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; ADR is inconsistent with the axiom of choice. It also implies the existence of inner models with certain large cardinals.

ADR is equivalent to AD plus the axiom of uniformization.

See also