In mathematics, a cardinal number is called huge if there exists an elementary embedding from into a transitive inner model with critical point and

Here, is the class of all sequences of length whose elements are in .

Huge cardinals were introduced by Kenneth Kunen (1978).

Variants

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In what follows,   refers to the  -th iterate of the elementary embedding  , that is,   composed with itself   times, for a finite ordinal  . Also,   is the class of all sequences of length less than   whose elements are in  . Notice that for the "super" versions,   should be less than  , not  .

κ is almost n-huge if and only if there is   with critical point   and

 

κ is super almost n-huge if and only if for every ordinal γ there is   with critical point  ,  , and

 

κ is n-huge if and only if there is   with critical point   and

 

κ is super n-huge if and only if for every ordinal   there is   with critical point  ,  , and

 

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  -huge for all finite  .

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.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named   through  , and a property  .[1] The additional property   is equivalent to "  is huge", and   is equivalent to "  is  -supercompact for all  ". Corazza introduced the property  , lying strictly between   and  .[2]

Consistency strength

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The cardinals are arranged in order of increasing consistency strength as follows:

  • almost  -huge
  • super almost  -huge
  •  -huge
  • super  -huge
  • almost  -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

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One can try defining an  -huge cardinal   as one such that an elementary embedding   from   into a transitive inner model   with critical point   and  , where   is the supremum of   for positive integers  . 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  -huge cardinal   is defined as the critical point of an elementary embedding from some rank   to itself. This is closely related to the rank-into-rank axiom I1.

See also

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References

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  1. ^ A. Kanamori, W. N. Reinhardt, R. Solovay, "Strong Axioms of Infinity and Elementary Embeddings", pp.110--111. Annals of Mathematical Logic vol. 13 (1978).
  2. ^ P. Corazza, "A new large cardinal and Laver sequences for extendibles", Fundamenta Mathematicae vol. 152 (1997).
  • Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3.
  • Kunen, Kenneth (1978), "Saturated ideals", The Journal of Symbolic Logic, 43 (1): 65–76, doi:10.2307/2271949, ISSN 0022-4812, JSTOR 2271949, MR 0495118, S2CID 13379542.
  • Maddy, Penelope (1988), "Believing the Axioms. II", The Journal of Symbolic Logic, 53 (3): 736-764 (esp. 754-756), doi:10.2307/2274569, JSTOR 2274569, S2CID 16544090. A copy of parts I and II of this article with corrections is available at the author's web page.