The first-order definability of generic large cardinals

TL;DR

This paper demonstrates that various notions of generic large cardinals, including supercompactness and hugeness, are first-order definable within the language of ZFC, extending to multiple large cardinal concepts.

Contribution

It establishes the first-order definability of generic large cardinal notions in ZFC, unifying several concepts under a common formal framework.

Findings

Generic supercompactness is first-order definable in ZFC.

First-order definability extends to generic hugeness and other large cardinals.

The results unify the understanding of generic large cardinals within formal logic.

Abstract

We show that the notions of generic and Laver-generic supercompactness are first-order definable in the language of ZFC. This also holds for generic and Laver-generic (almost) hugeness as well as for generic versions of other large cardinals.