View source for Huge cardinal

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

:<math>{}^{j(\kappa)}M \subset M.\!</math>

Here, ''<sup>&alpha;</sup>M'' is the class of all [[sequence]]s of length α whose elements are in M.

Huge cardinals were introduced by {{harvs|txt|authorlink=Kenneth Kunen|first=Kenneth |last=Kunen|year=1978}}.

== Variants ==
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>&lt;&alpha;</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>.

κ is '''almost n-huge''' if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and

:<math>{}^{<j^n(\kappa)}M \subset M.\!</math>

κ is '''super almost n-huge''' if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ&lt;j(κ), and

:<math>{}^{<j^n(\kappa)}M \subset M.\!</math>

κ is '''n-huge''' if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and

:<math>{}^{j^n(\kappa)}M \subset M.\!</math>

κ is '''super n-huge''' if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ&lt;j(κ), and

:<math>{}^{j^n(\kappa)}M \subset M.\!</math>

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''.

The existence of an almost huge cardinal implies that [[Vopěnka's principle]] is consistent; more precisely any almost huge cardinal is also a [[Vopěnka cardinal]].

== Consistency strength ==
The cardinals are arranged in order of increasing consistency strength as follows:
*almost ''n''-huge
*super almost ''n''-huge
*''n''-huge
*super ''n''-huge
*almost ''n''+1-huge
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).

==ω-huge cardinals==
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 such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF. Instead an ω-huge cardinal κ is defined as the critical point of an elementary embedding from some rank ''V''<sub>λ+1</sub> to itself. This is closely related to the [[rank-into-rank]] axiom I<sub>1</sub>.

== See also ==

*[[List of large cardinal properties]]
*The [[Dehornoy order]] on a braid group was motivated by properties of huge cardinals.

== References ==
*{{Citation|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd|isbn=3-540-00384-3}}.
*{{Citation | last=Kunen | first=Kenneth | authorlink=Kenneth Kunen | title=Saturated ideals | doi=10.2307/2271949 | jstor=2271949 | mr=495118 | year=1978 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=43 | issue=1 | pages=65–76}}.
*{{Citation|last=Maddy|first=Penelope|authorlink=Penelope Maddy|journal=The Journal of Symbolic Logic|title=Believing the Axioms. II|year=1988|volume=53|issue=3|pages=736-764 (esp. 754-756)|doi=10.2307/2274569|jstor=2274569}}. A copy of parts I and II of this article with corrections is available at the [http://faculty.sites.uci.edu/pjmaddy/bibliography/ author's web page].

[[Category:Large cardinals]]