Huge cardinal

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

[math]\displaystyle{ {}^{j(\kappa)}M \subset M. }[/math]

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

Huge cardinals were introduced by Kenneth Kunen (1978).

Variants

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

κ is almost n-huge if and only if there is [math]\displaystyle{ j : V \to M }[/math] with critical point [math]\displaystyle{ \kappa }[/math] and

[math]\displaystyle{ {}^{\lt j^n(\kappa)}M \subset M. }[/math]

κ is super almost n-huge if and only if for every ordinal γ there is [math]\displaystyle{ j : V \to M }[/math] with critical point [math]\displaystyle{ \kappa }[/math], [math]\displaystyle{ \gamma\lt j(\kappa) }[/math], and

[math]\displaystyle{ {}^{\lt j^n(\kappa)}M \subset M. }[/math]

κ is n-huge if and only if there is [math]\displaystyle{ j : V \to M }[/math] with critical point [math]\displaystyle{ \kappa }[/math] and

[math]\displaystyle{ {}^{j^n(\kappa)}M \subset M. }[/math]

κ is super n-huge if and only if for every ordinal [math]\displaystyle{ \gamma }[/math] there is [math]\displaystyle{ j : V \to M }[/math] with critical point [math]\displaystyle{ \kappa }[/math], [math]\displaystyle{ \gamma\lt j(\kappa) }[/math], and

[math]\displaystyle{ {}^{j^n(\kappa)}M \subset M. }[/math]

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 [math]\displaystyle{ n }[/math]-huge for all finite [math]\displaystyle{ n }[/math].

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 [math]\displaystyle{ \mathbf A_2(\kappa) }[/math] through [math]\displaystyle{ \mathbf A_7(\kappa) }[/math], and a property [math]\displaystyle{ \mathbf A_6^\ast(\kappa) }[/math]. The additional property [math]\displaystyle{ \mathbf A_1(\kappa) }[/math] is equivalent to "[math]\displaystyle{ \kappa }[/math] is huge", and [math]\displaystyle{ \mathbf A_3(\kappa) }[/math] is equivalent to "[math]\displaystyle{ \kappa }[/math] is [math]\displaystyle{ \lambda }[/math]-supercompact for all [math]\displaystyle{ \lambda\lt j(\kappa) }[/math]".

Consistency strength

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

• almost [math]\displaystyle{ n }[/math]-huge
• super almost [math]\displaystyle{ n }[/math]-huge
• [math]\displaystyle{ n }[/math]-huge
• super [math]\displaystyle{ n }[/math]-huge
• almost [math]\displaystyle{ n+1 }[/math]-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 [math]\displaystyle{ \omega }[/math]-huge cardinal [math]\displaystyle{ \kappa }[/math] as one such that an elementary embedding [math]\displaystyle{ j : V \to M }[/math] from [math]\displaystyle{ V }[/math] into a transitive inner model [math]\displaystyle{ M }[/math] with critical point [math]\displaystyle{ \kappa }[/math] and [math]\displaystyle{ {}^\lambda M\subseteq M }[/math], where [math]\displaystyle{ \lambda }[/math] is the supremum of [math]\displaystyle{ j^n(\kappa) }[/math] for positive integers [math]\displaystyle{ n }[/math]. 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 [math]\displaystyle{ \omega }[/math]-huge cardinal [math]\displaystyle{ \kappa }[/math] is defined as the critical point of an elementary embedding from some rank [math]\displaystyle{ V_{\lambda+1} }[/math] 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 .
• Maddy, Penelope (1988), "Believing the Axioms. II", The Journal of Symbolic Logic 53 (3): 736-764 (esp. 754-756), doi:10.2307/2274569 . A copy of parts I and II of this article with corrections is available at the author's web page.