Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka 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 $\omega$-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not ${\Delta }_2$-Mahlo" and $\text{gVP}({\mathbf{\Pi }}_1)$ + "there is no proper class of remarkable cardinals" are both equiconsistent with the existence of a proper class of $\omega$-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler.