In mathematics, a cardinal number is called huge if there exists an elementary embedding from into a transitive inner model with critical point and

Here, is the class of all sequences of length whose elements are in .

Huge cardinals were introduced by .

Variants In what follows, refers to the -th iterate of the elementary embedding , that is, composed with itself times, for a finite ordinal . Also, is the class of all sequences of length less than whose elements are in . Notice that for the "super" versions, should be less than , not .

κ is almost n-huge if and only if there is with critical point and

In particular, κ is said to be almost huge when it is almost 1-huge.

κ is super almost n-huge if and only if for every ordinal γ there is with critical point , , and

κ is n-huge if and only if there is with critical point and

κ is super n-huge if and only if for every ordinal there is with critical point , , 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 -huge for all finite .

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 through , and a property .A. Kanamori, W. N. Reinhardt, R. Solovay, "Strong Axioms of Infinity and Elementary Embeddings", pp.110--111. Annals of Mathematical Logic vol. 13 (1978). The additional property is equivalent to " is huge", and is equivalent to " is almost huge". Corazza introduced the property , lying strictly between and .P. Corazza, "A new large cardinal and Laver sequences for extendibles", Fundamenta Mathematicae vol. 152 (1997).

Consistency strength The cardinals are arranged in order of increasing consistency strength as follows: almost -huge super almost -huge -huge super -huge almost -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 -huge cardinal as one such that an elementary embedding from into a transitive inner model with critical point and , where is the supremum of for positive integers . 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 -huge cardinal is defined as the critical point of an elementary embedding from some rank 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

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