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

TL;DR

This paper explores weakly remarkable cardinals, their relation to Erdős cardinals, and their connection to the generic Vopěnka principle, establishing new consistency strength results and extending previous work.

Contribution

It introduces a weak version of remarkable cardinals, characterizes their reflection properties, and links their existence to Erdős cardinals and the generic Vopěnka principle.

Findings

Weakly remarkable cardinals can fail to be $oldsymbol{ m ext{ extSigma}_2}$-reflecting.

Existence of non-$oldsymbol{ m ext{ extSigma}_2}$-reflecting weakly remarkable cardinals is equiconsistent with $oldsymbol{ m ext{ extomega}- ext{ extErd ext{"o}s}}$ cardinals.

Certain forms of the generic Vopěnka principle are equiconsistent with the existence of a proper class of $oldsymbol{ m ext{ extomega}- ext{ extErd ext{"o}s}}$ cardinals.

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.