Generically extendible cardinals

In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of ${\omega }_1$ or ${\omega }_2$ has small consistency strength, but that of a cardinal $>{\omega }_2$ does not. We also consider some results concerned with generically extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic.