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| In [[mathematics]], a [[cardinal number]] κ is called '''huge''' if [[there exists]] an [[elementary embedding]] ''j'' : ''V'' → ''M'' from ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)|critical point]] κ and
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| :<math>{}^{j(\kappa)}M \subset M.\!</math>
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| Here, ''<sup>α</sup>M'' is the class of all [[sequence]]s of length α whose elements are in M.
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| Huge cardinals were introduced by {{harvs|txt|authorlink=Kenneth Kunen|first=Kenneth |last=Kunen|year=1978}}.
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| == Variants ==
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| In what follows, j<sup>''n''</sup> refers to the ''n''-th iterate of the elementary embedding j, that is, j [[function composition|composed]] with itself ''n'' times, for a finite ordinal ''n''. Also, ''<sup><α</sup>M'' is the class of all sequences of length less than α whose elements are in M. Notice that for the "super" versions, γ should be less than j(κ), not <math>{j^n(\kappa)}</math>.
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| κ is '''almost n-huge''' if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and
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| :<math>{}^{<j^n(\kappa)}M \subset M.\!</math>
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| κ is '''super almost n-huge''' if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ<j(κ), and
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| :<math>{}^{<j^n(\kappa)}M \subset M.\!</math>
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| κ is '''n-huge''' if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and
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| :<math>{}^{j^n(\kappa)}M \subset M.\!</math>
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| κ is '''super n-huge''' if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ<j(κ), and
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| :<math>{}^{j^n(\kappa)}M \subset M.\!</math>
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| Notice that 0-huge is the same as [[measurable cardinal]]; and 1-huge is the same as huge. A cardinal satisfying one of the [[rank into rank]] axioms is ''n''-huge for all finite ''n''.
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| The existence of an almost huge cardinal implies that [[Vopenka's principle]] is consistent; more precisely any almost huge cardinal is also a [[Vopenka cardinal]].
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| == Consistency strength ==
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| The cardinals are arranged in order of increasing consistency strength as follows:
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| *almost ''n''-huge
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| *super almost ''n''-huge
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| *''n''-huge
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| *super ''n''-huge
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| *almost ''n''+1-huge
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| The consistency of a huge cardinal implies the consistency of a [[supercompact cardinal]], nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).
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| ==ω-huge cardinals==
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| One can try defining an ω-huge cardinal κ as one such that an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and <sup>λ</sup>''M''⊆''M'', where λ is the supremum of ''j''<sup>''n''</sup>(κ) for positive integers ''n''. However [[Kunen's inconsistency theorem]] shows that ω-huge cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF.
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| == See also ==
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| *[[List of large cardinal properties]]
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| *The [[Dehornoy order]] on a braid group was motivated by properties of huge cardinals.
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| == References ==
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| *{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | title=Saturated ideals | doi=10.2307/2271949 | mr=495118 | year=1978 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=43 | issue=1 | pages=65–76}}
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| * Penelope Maddy,"Believing the Axioms,II"(i.e. part 2 of 2),"Journal of Symbolic Logic",vol.53,no.3,Sept.1988,pages 736 to 764 (esp.754-756).
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| * {{cite book|last=Kanamori|first=Akihiro|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd ed|isbn=3-540-00384-3}}
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| [[Category:Large cardinals]]
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Hi, everybody! My name is Kimberley.
It is a little about myself: I live in United States, my city of Datil.
It's called often Eastern or cultural capital of NM. I've married 4 years ago.
I have two children - a son (Florine) and the daughter (Sabrina). We all like Roller Derby.
Feel free to visit my site ... Fifa 15 Coin generator