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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
| | The author is known the particular name of Gabrielle Lattimer though she doesn't tremendously like being called like this. To bake is something that this lady has been doing for months and months. Her job is probably a cashier but inside the her [http://Photobucket.com/images/husband husband] and your will start their own home office. She's always loved living into South Carolina. She is running and examining a blog here: http://prometeu.net<br><br>Look into my homepage; [http://prometeu.net clash of clans hack 2014] |
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| :''f'' : λ → λ
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| there exists a cardinal κ < λ with
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| : {''f''(β)|β < κ} ⊆ κ
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| and an [[elementary embedding]]
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| :''j'' : ''V'' → ''M''
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| from the [[Von Neumann universe]] ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)|critical point]] κ and
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| :V<sub>j(f)(κ)</sub> ⊆ ''M''.
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| 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.
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| <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]].)
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| 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]].
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| == Consequences ==
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| 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).
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| 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)]]).
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| [[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.
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| 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>.
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| ==Hyper-Woodin cardinals==
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| 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
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| :{λ < κ | λ is <κ-''S''-[[strong cardinal|strong]]}
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| is in ''U''.
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| λ is <κ-S-strong if and only if for each δ < κ there is a [[transitive class]] ''N'' and an [[elementary embedding]]
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| :j : V → N | |
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| with
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| :λ = crit(j),
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| :j(λ)≥ δ, and
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| :<math>j(S) \cap H_\delta = S \cap H_\delta</math>.
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| The name alludes to the classical result that a cardinal is Woodin if and only if for every set ''S'', the set
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| :{λ < κ | λ is <κ-''S''-[[strong cardinal|strong]]}
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| is a [[stationary set]]
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| The measure ''U'' will contain the set of all [[Shelah cardinal]]s below κ.
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| ==Weakly hyper-Woodin cardinals==
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| 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
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| embedding j : V → N with λ = crit(j), j(λ) >= δ, and <math>j(S) \cap H_\delta = S \cap H_\delta.</math>
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| 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.
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| 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.
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| == Notes and references ==
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| <references/>
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| == Further reading ==
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| * {{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}}
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| * 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.
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| * Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, [http://www.math.cmu.edu/~eschimme/papers/hyperwoodin.pdf online]
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| * {{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 = 1146–7 | url = http://www.ams.org/notices/200709/tx070901146p.pdf | format = [[PDF]] | accessdate = 2008-01-15 }}
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| {{DEFAULTSORT:Woodin Cardinal}}
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| [[Category:Large cardinals]]
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| [[Category:Determinacy]]
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The author is known the particular name of Gabrielle Lattimer though she doesn't tremendously like being called like this. To bake is something that this lady has been doing for months and months. Her job is probably a cashier but inside the her husband and your will start their own home office. She's always loved living into South Carolina. She is running and examining a blog here: http://prometeu.net
Look into my homepage; clash of clans hack 2014