In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number
which is an elementary embedding with critical point
However, the first Woodin cardinal is not even weakly compact.
(known as the von Neumann hierarchy) is defined by transfinite recursion on
is no longer a set, but a proper class.
satisfies second-order ZFC ("satisfies" here means the notion of satisfaction from first-order logic).
is said to be an elementary embedding if for any formula
in the language of set theory, it is the case that
is first-order logic's notion of satisfaction as before.
is called nontrivial if it is not the identity.
is a nontrivial elementary embedding, there exists an ordinal
Many large cardinal properties can be phrased in terms of elementary embeddings.
can be found such that there is a nontrivial elementary embedding
[1] Woodin cardinals are important in descriptive set theory.
By a result[2] 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 (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.
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 (see Θ (set theory)).
Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a
[3] Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on
Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an
is called hyper-Woodin if there exists a normal measure
and an elementary embedding with The name alludes to the classical result that a cardinal is Woodin if and only if for every set
is called weakly hyper-Woodin if for every set
there exists a normal measure
The name alludes to the classic result that a cardinal is Woodin if for every set
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
be the least admissible ordinal greater than
These cardinals appear when building models from iteration trees.