Woodin Cardinal

A Woodin cardinal is a large cardinal, an infinite cardinal number whose existence implies a strong and highly structured form of infinity going well beyond what the standard Zermelo-Fraenkel axioms of set theory can prove, defined through a technical condition involving elementary embeddings that must hold for every function on the cardinal. It is named after the American mathematician W. Hugh Woodin, who introduced the notion in the 1980s while working on questions connecting large cardinal axioms to the deeper structure of sets of real numbers. The existence of infinitely many Woodin cardinals is known to imply projective determinacy, the striking and previously unprovable statement that every two player game of a certain broad definable class, played by choosing real numbers according to fixed rules, has a winning strategy for one of the two players, a result that ties an axiom about very large infinite cardinals directly to concrete questions in descriptive set theory. Because of this connection, Woodin cardinals occupy a central place in the modern study of the foundations of mathematics and the search for further axioms to extend standard set theory.