Huge cardinal

In mathematics, a cardinal number

is called huge if there exists an elementary embedding

into a transitive inner model

with critical point

is the class of all sequences of length

Huge cardinals were introduced by Kenneth Kunen (1978).

-th iterate of the elementary embedding

times, for a finite ordinal

is the class of all sequences of length less than

Notice that for the "super" versions,

κ is almost n-huge if and only if there is

with critical point

and κ is super almost n-huge if and only if for every ordinal γ there is

with critical point

γ < j ( κ )

, and κ is n-huge if and only if there is

and κ is super n-huge if and only if for every ordinal

, 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

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

λ < j ( κ )

Corazza introduced the property

, lying strictly between

[2] The cardinals are arranged in order of increasing consistency strength as follows: 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).

as one such that an elementary embedding

into a transitive inner model

However Kunen's inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF.

is defined as the critical point of an elementary embedding from some rank

This is closely related to the rank-into-rank axiom I1.