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 κ and

${\displaystyle {}^{j(\kappa )}M\subset M.\!}$

Here, ^αM 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, jⁿ 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, ^<αM 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 ${\displaystyle {j^{n}(\kappa )}}$.

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

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

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

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

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

${\displaystyle {}^{j^{n}(\kappa )}M\subset M.\!}$

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

${\displaystyle {}^{j^{n}(\kappa )}M\subset M.\!}$

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.

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 ^λM⊆M, where λ is the supremum of jⁿ(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 I₁.

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.