In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ such that for all functions
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f : λ → λthere exists a cardinal κ < λ with
{f(β)|β < κ} ⊆ κand an elementary embedding
j : V → Mfrom the Von Neumann universe 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
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 (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
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω1 is
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
{λ < κ | λ is <κ-S-strong}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 → Nwith
λ = crit(j),j(λ)≥ δ, andThe name alludes to the classical result that a cardinal is Woodin if and only if for every set S, the set
{λ < κ | λ is <κ-S-strong}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 {λ < κ | λ is <κ-S-strong} 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), j(λ) >= δ, and
The name alludes to the classic result that a cardinal is Woodin if for every set S, the set {λ < κ | λ is <κ-S-strong} 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.