Huge cardinal explained

\kappa is called huge if there exists an elementary embedding j : V \to M from V into a transitive inner model M with critical point \kappa and{} ^{j(\kappa)} M \subset M.

Here,

{} ^{\alpha} M is the class of all sequences of length \alpha whose elements are in M .

Huge cardinals were introduced by .

Variants

In what follows,

j ^{n} 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, {} ^{<\alpha} M is the class of all sequences of length less than \alpha whose elements are in M . Notice that for the "super" versions, \gamma should be less than j(\kappa) , not {j ^{n(\kappa)}} .

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

j : V \to M with critical point \kappa and{} ^{<j ^{n(\kappa)}} M \subset M.

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

j : V \to M with critical point \kappa , \gamma< j(\kappa) , and{} ^{<j ^{n(\kappa)}} M \subset M.

κ is n-huge if and only if there is

j : V \to M with critical point \kappa and{} ^{j ^{n(\kappa)}} M \subset M.

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

\gamma there is j : V \to M with critical point \kappa , \gamma< j(\kappa) , and{} ^{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.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named

A _{2(\kappa)} through A _{7(\kappa)} , and a property A ^{\ast(\kappa)} _{6} .^[1] The additional property A _{1(\kappa)} is equivalent to "\kappa is huge", and A _{3(\kappa)} is equivalent to "\kappa is λ -supercompact for all λ <j(\kappa) ". Corazza introduced the property A _{3.5} , lying strictly between A _{3} and A _{4} .^[2]

Consistency strength

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

• almost

n -huge

• super almost

n -hugen -huge

• super

n -huge

• almost

n +1 -hugeThe 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

\omega -huge cardinal \kappa as one such that an elementary embedding j : V \to M from V into a transitive inner model M with critical point \kappa and {} ^{λ} M\subseteq M , where λ is the supremum of j ^{n(\kappa)} 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 \omega -huge cardinal \kappa 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

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

Notes and References

1. A. Kanamori, W. N. Reinhardt, R. Solovay, "Strong Axioms of Infinity and Elementary Embeddings", pp.110--111. Annals of Mathematical Logic vol. 13 (1978).
2. P. Corazza, "A new large cardinal and Laver sequences for extendibles", Fundamenta Mathematicae vol. 152 (1997).