Huge cardinal

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

j ( κ ) M ⊂ M . {\displaystyle {}^{j(\kappa )}M\subset M.}

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

Huge cardinals were introduced by Kenneth Kunen (1978).

## Variants

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

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

< j n ( κ ) M ⊂ M . {\displaystyle {}^{<j^{n}(\kappa )}M\subset M.}

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

< j n ( κ ) M ⊂ M . {\displaystyle {}^{<j^{n}(\kappa )}M\subset M.}

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

j n ( κ ) M ⊂ M . {\displaystyle {}^{j^{n}(\kappa )}M\subset M.}

κ is super n-huge if and only if for every ordinal  γ {\displaystyle \gamma } there is  j : V -> M {\displaystyle j:V\to M} with critical point  κ {\displaystyle \kappa } ,  γ < j ( κ ) {\displaystyle \gamma <j(\kappa )} , and

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

## Consistency strength

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

* almost  n {\displaystyle n} -huge
* super almost  n {\displaystyle n} -huge
* n {\displaystyle n} -huge
* super  n {\displaystyle n} -huge
* almost  n + 1 {\displaystyle n+1} -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

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One can try defining an  ω {\displaystyle \omega } -huge cardinal  κ {\displaystyle \kappa } as one such that an elementary embedding  j : V -> M {\displaystyle j:V\to M} from  V {\displaystyle V} into a transitive inner model  M {\displaystyle M} with critical point  κ {\displaystyle \kappa } and  λ M ⊆ M {\displaystyle {}^{\lambda }M\subseteq M} , where  λ {\displaystyle \lambda } is the supremum of  j n ( κ ) {\displaystyle j^{n}(\kappa )} for positive integers  n {\displaystyle 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  ω {\displaystyle \omega } -huge cardinal  κ {\displaystyle \kappa } is defined as the critical point of an elementary embedding from some rank  V λ + 1 {\displaystyle V_{\lambda +1}} to itself. This is closely related to the rank-into-rank axiom I1.

## See also

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* List of large cardinal properties
* The Dehornoy order on a braid group was motivated by properties of huge cardinals.

## References

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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).

* 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, JSTOR 2271949, MR 0495118, S2CID 13379542.
* Maddy, Penelope (1988), "Believing the Axioms. II", The Journal of Symbolic Logic, 53 (3): 736-764 (esp. 754-756), doi:10.2307/2274569, JSTOR 2274569, S2CID 16544090. A copy of parts I and II of this article with corrections is available at the author's web page.