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.

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: 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).
 * almost $$n$$-huge
 * super almost $$n$$-huge
 * $$n$$-huge
 * super $$n$$-huge
 * almost $$n+1$$-huge

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.