Huge Cardinal - Variants

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 .

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

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

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 Vopenka's principle is consistent; more precisely any almost huge cardinal is also a Vopenka cardinal.