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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 &rarr; λ
 
there exists a cardinal κ < λ with
 
: {''f''(β)|β < κ} &sube; κ
 
and an [[elementary embedding]]
 
:''j'' : ''V'' &rarr; ''M''
 
from the [[Von Neumann universe]] ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)|critical point]] κ and
 
:V<sub>j(f)(κ)</sub> &sube; ''M''.
 
An equivalent definition is this: λ is Woodin [[if and only if]] λ is [[inaccessible cardinal|strongly inaccessible]] and for all <math>A \subseteq V_\lambda</math> there exists a <math>\lambda_A</math> < λ which is <math><\lambda</math>-<math>A</math>-strong.
 
<math>\lambda _A</math> being <math><\lambda</math>-<math>A</math>-strong means that for all [[ordinal number|ordinals]] α < λ, there exist a <math>j: V \to M</math> which is an [[elementary embedding]] with [[critical point (set theory)|critical point]] <math>\lambda _A</math>, <math>j(\lambda _A) > \alpha</math>, <math>V_\alpha \subseteq M</math> and <math>j(A) \cap V_\alpha = A \cap V_\alpha</math>. (See also [[strong cardinal]].)
 
A Woodin cardinal is preceded by a [[stationary set]] of [[measurable cardinal]]s, and thus it is a [[Mahlo cardinal]]. However, the first Woodin cardinal is not even [[Weakly compact cardinal|weakly compact]].
 
== Consequences ==
 
Woodin cardinals are important in [[descriptive set theory]].  By a result<ref>[http://www.jstor.org/stable/1990913 A Proof of Projective Determinacy]</ref> of [[Donald A. Martin|Martin]] and [[John R. Steel|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 [[meagre set|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 set|perfect]] subset).
 
The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses.  Working in [[Zermelo–Fraenkel set theory|ZF]]+[[axiom of determinacy|AD]]+[[axiom of dependent choice|DC]] one can prove that <math>\Theta _0</math> is Woodin in the class of hereditarily ordinal-definable sets. <math>\Theta _0</math> is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see [[Θ (set theory)]]).
 
[[Saharon Shelah|Shelah]] proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω<sub>1</sub> is <math>\aleph_2</math>-saturated.  
Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an <math>\aleph_1</math>-dense ideal over <math>\aleph_1</math>.
 
==Hyper-Woodin cardinals==
A [[cardinal number|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 cardinal|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)&ge; δ, and
 
:<math>j(S) \cap H_\delta = S \cap H_\delta</math>.
 
The 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 cardinal|strong]]}
 
is a [[stationary set]]
 
The measure ''U'' will contain the set of all [[Shelah cardinal]]s below κ.
 
==Weakly hyper-Woodin cardinals==
A [[cardinal number|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 <math>j(S) \cap H_\delta = S \cap H_\delta.</math>
 
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 cardinal|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.
 
== Notes and references ==
<references/>
 
== Further reading ==
* {{cite book|last=Kanamori|first=Akihiro|year=2003|authorlink=Akihiro Kanamori|publisher=Springer|title=The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings|edition=2nd|isbn=3-540-00384-3}}
 
* For proofs of the two results listed in consequences see ''Handbook of Set Theory'' (Eds. Foreman, Kanamori, Magidor) (to appear).  [http://handbook.assafrinot.com/ Drafts] of some chapters are available.
* Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp.&nbsp;3385–3391, 2002, [http://www.math.cmu.edu/~eschimme/papers/hyperwoodin.pdf online]
 
* {{cite journal | last = Steel | first = John R. | authorlink = John R. Steel |date=October 2007 | title = What is a Woodin Cardinal? | journal = [[Notices of the American Mathematical Society]] | volume = 54 | issue = 9 | pages = 1146&ndash;7 | url = http://www.ams.org/notices/200709/tx070901146p.pdf | format = [[PDF]] | accessdate = 2008-01-15 }}
 
{{DEFAULTSORT:Woodin Cardinal}}
[[Category:Large cardinals]]
[[Category:Determinacy]]

Latest revision as of 21:33, 25 June 2014

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