# Axiom of real determinacy Information

*https://en.wikipedia.org/wiki/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.

## See also