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
- Vj(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.
Read more about Woodin Cardinal: Consequences, Hyper-Woodin Cardinals, Weakly Hyper-Woodin Cardinals
Famous quotes containing the word cardinal:
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—W.H. (Wystan Hugh)