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

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

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

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

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

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

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

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

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

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 ${\displaystyle n}$ -huge for all finite ${\displaystyle n}$ .

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

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

• almost ${\displaystyle n}$ -huge
• super almost ${\displaystyle n}$ -huge
• ${\displaystyle n}$ -huge
• super ${\displaystyle n}$ -huge
• almost ${\displaystyle n+1}$ -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).

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

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