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.
Categories: math.LO