Editing Axiom of determinacy (section)

== Large cardinals and the axiom of determinacy ==
The consistency of the axiom of determinacy is closely related to the question of the consistency of [[large cardinal]] axioms. By a theorem of [[W. Hugh Woodin|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 cardinal]]s. Since Woodin cardinals are [[inaccessible cardinal|strongly inaccessible]], if AD is consistent, then so are an infinity of inaccessible cardinals.

Moreover, if to the hypothesis of an infinite set of Woodin cardinals is added the existence of a [[measurable cardinal]] larger than all of them, a very strong theory of [[Lebesgue measurable]] sets of reals emerges, as it is then provable that the axiom of determinacy is true in [[L(R)]], and therefore that ''every'' set of real numbers in L(R) is determined.