AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.
Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers.
Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis.
Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable.
Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory.
In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research.
Assuming the existence of some uncountable cardinal numbers analogous to ℵ0, they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
The axiom of determinacy refers to games of the following specific form: Consider a subset A of the Baire space ωω of all infinite sequences of natural numbers.
Two players alternately pick natural numbers That generates the sequence ⟨ni⟩i∈ω after infinitely many moves.
Define the function f: ωω → {0, 1}ω such that f(r) is the unique sequence of length ω with values are in {0, 1} whose first term equals 0, and whose sequence of runs (see run-length encoding) equals r. (Such an f can be shown to be injective.
In other words, we are choosing "half" of the elements of T, a subset that we denote by U (where U ⊆ T) such that t ∈ U iff h(t) ∉ U.
However, f(r) and f(r') differ in all but the first term (by the nature of run-length encoding and an offset of 1), so f(r) and f(r') are in complement equivalent classes, so g(f(r)), g(f(r')) cannot both be in U, contradicting the assumption that q is a winning strategy.
However, f(r) and f(r') differ in all but the first a1 + 1 terms, so they are in complement equivalent classes, therefore g(f(r)) and g(f(r')) cannot both be in U, contradicting that p is a winning strategy.
By a theorem of Woodin, the consistency of Zermelo–Fraenkel set theory without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many Woodin cardinals.
Assuming AD, all δ1n are initial ordinals, and we have δ12n+2 = (δ12n+1)+, and for n < ω, the 2n-th Suslin cardinal is equal to δ12n−1.