Woodin cardinal


In set theory, a Woodin cardinal is a cardinal number λ such that for all functions
there exists a cardinal κ < λ with
and an elementary embedding
from the Von Neumann universe V into a transitive inner model M with critical point κ and
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.
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.

Consequences

Woodin cardinals are important in descriptive set theory. By a result of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has the Baire property, and the perfect set property.
The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that is Woodin in the class of hereditarily ordinal-definable sets. is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection.
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω1 is -saturated.
Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an -dense ideal over.

Hyper-Woodin cardinals

A cardinal κ is called hyper-Woodin if there exists a normal measure U on κ such that for every set S, the set
is in U.
λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary embedding
with
The name alludes to the classical result that a cardinal is Woodin if and only if for every set S, the set
is a stationary set
The measure U will contain the set of all Shelah cardinals below κ.

Weakly hyper-Woodin cardinals

A cardinal κ is called weakly hyper-Woodin if for every set S there exists a normal measure U on κ such that the set is in U. λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary
embedding j : V → N with λ = crit, j >= δ, and
The name alludes to the classic result that a cardinal is Woodin if for every set S, the set is stationary.
The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of U does not depend on the choice of the set S for hyper-Woodin cardinals.