Huge cardinal

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Large cardinal from set theory

title: "Huge cardinal" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["large-cardinals"] description: "Large cardinal from set theory" topic_path: "general/large-cardinals" source: "https://en.wikipedia.org/wiki/Huge_cardinal" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Large cardinal from set theory ::

In mathematics, a cardinal number \kappa is called huge if there exists an elementary embedding j : V \to M from V into a transitive inner model M with critical point \kappa and

:{}^{j(\kappa)}M \subset M.

Here, {}^\alpha M is the class of all sequences of length \alpha whose elements are in M.

Huge cardinals were introduced by .

Variants

In what follows, j^n refers to the n-th iterate of the elementary embedding j, that is, j composed with itself n times, for a finite ordinal n. Also, {}^{ is the class of all sequences of length less than \alpha whose elements are in M. Notice that for the "super" versions, \gamma should be less than j(\kappa), not {j^n(\kappa)}.

κ is almost n-huge if and only if there is j : V \to M with critical point \kappa and

:{}^{

κ is super almost n-huge if and only if for every ordinal γ there is j : V \to M with critical point \kappa, \gamma, and

:{}^{

κ is n-huge if and only if there is j : V \to M with critical point \kappa and

:{}^{j^n(\kappa)}M \subset M.

κ is super n-huge if and only if for every ordinal \gamma there is j : V \to M with critical point \kappa, \gamma, and

:{}^{j^n(\kappa)}M \subset M.

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.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named \mathbf A_2(\kappa) through \mathbf A_7(\kappa), and a property \mathbf A_6^\ast(\kappa). The additional property \mathbf A_1(\kappa) is equivalent to "\kappa is huge", and \mathbf A_3(\kappa) is equivalent to "\kappa is \lambda-supercompact for all \lambda". Corazza introduced the property A_{3.5}, lying strictly between A_3 and A_4.

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 \omega-huge cardinal \kappa as one such that an elementary embedding j : V \to M from V into a transitive inner model M with critical point \kappa and {}^\lambda M\subseteq M, where \lambda is the supremum of j^n(\kappa) 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 \omega-huge cardinal \kappa is defined as the critical point of an elementary embedding from some rank V_{\lambda+1} to itself. This is closely related to the rank-into-rank axiom I1.

References

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  • . A copy of parts I and II of this article with corrections is available at the author's web page.

References

  1. A. Kanamori, W. N. Reinhardt, R. Solovay, "[https://math.bu.edu/people/aki/d.pdf Strong Axioms of Infinity and Elementary Embeddings]", pp.110--111. Annals of Mathematical Logic vol. 13 (1978).
  2. P. Corazza, "[http://matwbn.icm.edu.pl/ksiazki/fm/fm152/fm15225.pdf A new large cardinal and Laver sequences for extendibles]", Fundamenta Mathematicae vol. 152 (1997).

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large-cardinals