ⓘ Woodin cardinal
In set theory, a Woodin cardinal is a cardinal number λ such that for all functions
f: λ → λthere exists a cardinal κ < λ with
{ f ββ < κ} ⊆ κand an elementary embedding
j: V → Mfrom the Von Neumann universe V into a transitive inner model M with critical point κ and
V jfκ ⊆ M.An equivalent definition is this: λ is Woodin if and only if λ is strongly inaccessible and for all A ⊆ V λ {\displaystyle A\subseteq V_{\lambda }} there exists a λ A {\displaystyle \lambda _{A}} < λ which is < λ {\displaystyle α {\displaystyle j\lambda _{A}> \alpha }, V α ⊆ M {\displaystyle V_{\alpha }\subseteq M} and j A ∩ V α = A ∩ V α {\displaystyle jA\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. However, the first Woodin cardinal is not even weakly compact.
1. 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 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 Θ 0 {\displaystyle \Theta _{0}} is Woodin in the class of hereditarily ordinaldefinable sets. Θ 0 {\displaystyle \Theta _{0}} is the first ordinal onto which the continuum cannot be mapped by an ordinaldefinable 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 ω 1 is ℵ 2 {\displaystyle \aleph _{2}} saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an ℵ 1 {\displaystyle \aleph _{1}} dense ideal over ℵ 1 {\displaystyle \aleph _{1}}.
 models and determinacy. A type of large cardinal the Woodin cardinal bears his name. Born in Tucson, Arizona, Woodin earned his Ph.D. from the University
 New York politician William Hartman Woodin 1868  1934 US Secretary of Treasury Woodin, Arizona, a populated place in Coconino County Woodin cardinal
 Ramsey cardinals measurable cardinals 0 λ  strong, strong cardinals tall cardinals Woodin weakly hyper  Woodin Shelah, hyper  Woodin cardinals superstrong
 measurable limit λ of both Woodin cardinals and cardinals strong up to λ. If V has Woodin cardinals but not cardinals strong past a Woodin one, then under appropriate
 Shelah cardinal has a normal ultrafilter containing the set of weakly hyper  Woodin cardinals below it. Ernest Schimmerling, Woodin cardinals Shelah
 Woodin cardinal Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal
 and Woodin cardinals in consistency strength. However, the least strong cardinal is larger than the least superstrong cardinal Every strong cardinal is
 Woodin W. Hugh 2001 The continuum hypothesis, part II Notices of the American Mathematical Society. 48 7 681 690. Large Cardinals and Determinacy
 Ramsey cardinal Erdos cardinal Extendible cardinal Huge cardinal Hyper  Woodin cardinal Inaccessible cardinal Ineffable cardinal Mahlo cardinal Measurable
 there is a Woodin cardinal with a measurable cardinal above it, then Π12 determinacy holds. More generally, if there are n Woodin cardinals with a measurable

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