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

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

Huge cardinals were introduced by Kenneth Kunen (1978).

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

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

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

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