  The generic Vop\v{e}nka principle, we prove, is relatively consistent with
the ordinals being non-Mahlo. Similarly, the generic Vop\v{e}nka scheme is
relatively consistent with the ordinals being definably non-Mahlo. Indeed, the
generic Vop\v{e}nka scheme is relatively consistent with the existence of a
$\Delta_2$-definable class containing no regular cardinals. In such a model,
there can be no $\Sigma_2$-reflecting cardinals and hence also no remarkable
cardinals. This latter fact answers negatively a question of Bagaria, Gitman
and Schindler.

Logic