nLab
C(n)-extendible cardinal

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 $<\kappa$.

Definition

If $\kappa <\lambda$ 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 $\mu \in C(n)$, $j(\kappa )>\lambda$, 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 $\lambda >\kappa$.

In particular:

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

References

• Joan Bagaria, Carles Casacuberta, Adrian Mathias, Jiri Rosicky Definable orthogonality classes in accessible categories are small, arXiv