  In this paper, we study generically extendible cardinal, which is a generic
version of extendible cardinal. We prove that the generic extendibility of
$\omega_1$ or $\omega_2$ has small consistency strength, but of a cardinal
$>\omega_2$ is not. We also consider some results concerning with generic
extendible cardinals, such as indestructibility, generic absoluteness of the
reals, and Boolean valued second order logic.
