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 $\mathbf {A} _{6}^{\ast }(\kappa )$. 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 $A_{4}$.

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.

See also

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

References

• 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.