Woodin Cardinal

In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ such that for all functions

f : λ → λ

there exists a cardinal κ < λ with

{f(β)|β < κ} ⊆ κ

and an elementary embedding

j : V → M

from V into a transitive inner model M with critical point κ and

V_j(f)(κ) ⊆ M.

An equivalent definition is this: λ is Woodin if and only if λ is strongly inaccessible and for all there exists a < λ which is --strong.

being --strong means that for all ordinals α < λ, there exist a which is an elementary embedding with critical point, and . (See also strong cardinal.)

A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.