  We consider a weak version of Schindler's remarkable cardinals that may fail
to be $\Sigma_2$-reflecting. We show that the $\Sigma_2$-reflecting weakly
remarkable cardinals are exactly the remarkable cardinals, and we show that the
existence of a non-$\Sigma_2$-reflecting weakly remarkable cardinal has higher
consistency strength: it is equiconsistent with the existence of an
$\omega$-Erd\H{o}s cardinal. We give an application involving gVP, the generic
Vop\v{e}nka principle defined by Bagaria, Gitman, and Schindler. Namely, we
show that gVP + "Ord is not $\Delta_2$-Mahlo" and $\text{gVP}({\bf\Pi}_1)$ +
"there is no proper class of remarkable cardinals" are both equiconsistent with
the existence of a proper class of $\omega$-Erd\H{o}s cardinals, extending
results of Bagaria, Gitman, Hamkins, and Schindler.
