Home > Woodin cardinal

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 the 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 the perfect subset property (is either countable or contains a perfect subset).