弱显着基数、Erdés基数和泛伏普尼卡原理
Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle
摘要：
我们认为辛德勒的显着的红衣主教，可能无法是$Σ_2$ -反映弱版本。我们证明了$Σ_2$ -反射弱显著基数是显著基数，并且证明了非$Σ_2$ -反射弱显著基数的存在性具有更高的一致性强度：它与$ω$ -Erdés基数的存在性是等一致的.我们给出了一个应用程序，涉及GVP，一般Vopponka原则定义的Bagaria，Gitman，和辛德勒。也就是说，我们证明了gVP +“Ord不是$Δ_2$ -Mahlo”和$\text{gVP}({\bfΠ}_1)$ +“没有适当的显着基数类”都与存在适当的$ω$ -Erdibs基数类等相容，推广了Bagaria，Gitman，Hamkins和Schindler的结果.
英文摘要：
We consider a weak version of Schindler's remarkable cardinals that may fail to be $Σ_2$-reflecting. We show that the $Σ_2$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-$Σ_2$-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 $Δ_2$-Mahlo" and $\text{gVP}({\bfΠ}_1)$ + "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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