Woodin cardinal
In set theory, a Woodin cardinal is a cardinal number such that for all functions, there exists a cardinal with and an elementary embedding from the Von Neumann universe into a transitive inner model with critical point and.
An equivalent definition is this: is Woodin if and only if is strongly inaccessible and for all there exists a which is --strong.
being --strong means that for all ordinals, 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.p. 364
Explanation
The hierarchy is defined by transfinite recursion on :- ,
- ,
- , when is a limit ordinal.
For a transitive class, a function is said to be an elementary embedding if for any formula with free variables in the language of set theory, it is the case that iff, where is first-order logic's notion of satisfaction as before. An elementary embedding is called nontrivial if it is not the identity. If is a nontrivial elementary embedding, there exists an ordinal such that, and the least such is called the critical point of.
Many large cardinal properties can be phrased in terms of elementary embeddings. For an ordinal, a cardinal is said to be -strong if a transitive class can be found such that there is a nontrivial elementary embedding whose critical point is, and in addition.
A strengthening of the notion of -strong cardinal is the notion of -strongness of a cardinal in a greater cardinal : if and are cardinals with, and is a subset of, then is said to be -strong in if for all, there is a nontrivial elementary embedding witnessing that is -strong, and in addition. Finally, a cardinal is Woodin if for any choice of, there exists a such that is -strong in.
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 Lebesgue 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.
Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a -well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds.
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on 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 is called hyper-Woodin if there exists a normal measure on such that for every set, the setis in.
is --strong if and only if for each there is a transitive class and an elementary embedding
with
The name alludes to the classical result that a cardinal is Woodin if and only if for every set, the set
is a stationary set.p. 363
The measure will contain the set of all Shelah cardinals below.
Weakly hyper-Woodin cardinals
A cardinal is called weakly hyper-Woodin if for every set there exists a normal measure on such that the set is --strong is in. is --strong if and only if for each there is a transitive class and an elementaryembedding with,, and p. 3390
The name alludes to the classic result that a cardinal is Woodin if for every set, the set is --strong is stationary.
The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of does not depend on the choice of the set for hyper-Woodin cardinals.