Woodin cardinal

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

:"f" : λ → λ

there exists

:κ < λ with {"f"(β)|β<κ} &sube; κ

and an elementary embedding

:"j" : "V" &rarr; "M"

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

:V_j(f)(κ) &sube; "M".

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

$$lambda _A being -$$A-strong means that for all ordinals α < λ, there exist a $$j: V o M which is an elementary embedding with critical point $$lambda _A, $$j(lambda _A) > alpha, $$V_alpha subseteq M and $$j(A) cap V_alpha = A cap V_alpha. (See also strong cardinal.)

A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. It need not be measurable, however.

Consequences

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

The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that $$Theta _0 is Woodin in the class of hereditarily ordinal-definable sets. $$Theta _0 is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)).

Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on &omega;₁ is $$aleph_2-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an $$aleph_1-dense ideal over $$aleph_1.

References

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* For proofs of the two results listed in consequences see "Handbook of Set Theory" (Eds. Foreman, Kanamori, Magidor) (to appear). [http://www.tau.ac.il/~rinot/host.html Drafts] of some chapters are available.

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