Weakly remarkable cardinals, Erd\H{o}s cardinals, and the generic Vop\v{e}nka principle

Abstract

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.