# Huge, almost huge and superhuge cardinals

A cardinal $\kappa$ is huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)=\lambda$ and $M^\lambda\subset M$.

## Almost huge

A cardinal $\kappa$ is almost huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)=\lambda$ and $M^{\lt\lambda}\subset M$.

## Superhuge

A cardinal $\kappa$ is superhuge if it is huge with target $\lambda$ for unboundedly many cardinals $\lambda$.

## n huge

A cardinal $\kappa$ is $n$-huge with target $\lambda$ if there is an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M^{\kappa_n}\subset M$, where $\kappa_0=\kappa$ and $k_{i+1}=j(\kappa_i)$. Thus, the $1$-huge cardinals are precisely the huge cardinals.

## Super n huge

A cardinal $\kappa$ is super-$n$-huge if it is $n$-huge with target $\lambda$ for arbitrarily large cardinals $\lambda$.

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