| Summary: | We consider a weak version of Schindler's remarkable cardinals that may fail to be Σ2 - reflecting. We show that the Σ2-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-Σ2 -reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdos 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 Δ2-Mahlo" and gVP(Π1) + "there is no proper class of remarkable cardinals" are both equiconsistent with the existence of a proper class of ω-Erdos cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. Copyright © 2019 The Association for Symbolic Logic.
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