Woodin cardinal

A cardinal number κ is called a Woodin cardinal iff for all f : κ → κ there exists α < κ with f[α] ⊆ α and an elementary embedding j : V → M from V into a transitive inner model M with critical point α and V_j(f)(α) ⊆ M.

Woodin cardinals are important in descriptive set theory. Existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has Baire property (differs from an open set by a meager set, that is a set which is a countable union of nowhere dense sets), and perfect subset property (is either countable or contains a perfect subset).