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 txt.

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, {}^{<\alpha}M 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

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

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< j(\kappa), and

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

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< j(\kappa), 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).[1] 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<j(\kappa)". Corazza introduced the property A_{3.5}, lying strictly between A_3 and A_4.[2]

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).[3][4]

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.

See also

References

  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).
  3. ^ Asaf Karagila (2025). "Large Cardinals"
  4. ^ Morgenstern, Carl (1979), "On the ordering of certain large cardinals", Journal of Symbolic Logic. 44 (4).