Huge cardinal

In mathematics, a cardinal number $\kappa$ 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 Kenneth Kunen (1978).

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 $\mathbf {A} _{2}(\kappa )$ through $\mathbf {A} _{7}(\kappa )$ , and a property $\mathbf {A} _{6}^{\ast }(\kappa )$ .^[1] The additional property $\mathbf {A} _{1}(\kappa )$ is equivalent to " $\kappa$ is huge", and $\mathbf {A} _{3}(\kappa )$ is equivalent to " $\kappa$ is $\lambda$ -supercompact for all $\lambda <j(\kappa )$ ". 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 $n$ -huge
$n$ -huge
super $n$ -huge
almost $n+1$ -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

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 ${}^{\lambda }M\subseteq M$ , where $\lambda$ 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 $V_{\lambda +1}$ to itself. This is closely related to the rank-into-rank axiom I₁.

References

^ A. Kanamori, W. N. Reinhardt, R. Solovay, "Strong Axioms of Infinity and Elementary Embeddings", pp.110--111. Annals of Mathematical Logic vol. 13 (1978).
^ 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.

[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).

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