Weakly remarkable cardinals, erdős cardinals, and the generic vopěnka principle

Journal of Symbolic Logic 84 (4):1711-1721 (2019)
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Abstract

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-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 ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler.

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References found in this work

Strong axioms of infinity and elementary embeddings.Robert M. Solovay - 1978 - Annals of Mathematical Logic 13 (1):73.
C(n)-cardinals.Joan Bagaria - 2012 - Archive for Mathematical Logic 51 (3-4):213-240.
Virtual large cardinals.Victoria Gitman & Ralf Schindler - 2018 - Annals of Pure and Applied Logic 169 (12):1317-1334.
Proper forcing and remarkable cardinals.Ralf-Dieter Schindler - 2000 - Bulletin of Symbolic Logic 6 (2):176-184.

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