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 κ and

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

Huge cardinals were introduced by Kenneth Kunen (1978).

## 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, * ^{<α}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 .

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

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

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

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

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 ^{λ}*M*⊆*M*, where λ is the supremum of *j*^{n}(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*_{λ+1} to itself. This is closely related to the rank-into-rank axiom I_{1}.

## See also

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

## References

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