Huge cardinal

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

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

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).