# $C(n)$-extendible cardinals

## Idea

A $C(n)$-extendible cardinal is a type of large cardinal which generalizes extendible cardinals. They come up in formulating refined versions of Vopěnka's principle.

## Definition

We work in ZFC. Let $C(n)$ denote the class of infinite cardinal numbers $\kappa$ such that $V_\kappa$ (see von Neumann hierarchy) is a $\Sigma_n$-elementary submodel of the universe $V$, i.e. such that the embedding preserves and reflects the truth of $\Sigma_n$-formulas. In particular:

• $\kappa$ is $C(0)$ if and only if it is infinite, and

• $\kappa$ is $C(1)$ if and only if it is uncountable and $V_\kappa$ coincides with the hereditarily $\kappa$-sized sets $H(\kappa)$, i.e. those whose transitive closure has cardinality $\lt\kappa$.

###### Definition

If $\kappa\lt\lambda$ and both are in $C(n)$, we say $\kappa$ is $\lambda$-$C(n)$-extendible if there is an elementary embedding $j:V_\lambda \to V_\mu$ with critical point $\kappa$, such that $\mu\in C(n)$, $j(\kappa)\gt \lambda$, and $j(\kappa)\in C(n)$.

We say $\kappa\in C(n)$ is $C(n)$-extendible if it is $\lambda$-$C(n)$-extendible for all $\lambda\in C(n)$ with $\lambda\gt\kappa$.

In particular:

• $\kappa$ is $C(0)$-extendible if and only if it is extendible.