1 results
WEAKLY REMARKABLE CARDINALS, ERDŐS CARDINALS, AND THE GENERIC VOPĚNKA PRINCIPLE
-
- Journal:
- The Journal of Symbolic Logic / Volume 84 / Issue 4 / December 2019
- Published online by Cambridge University Press:
- 13 September 2019, pp. 1711-1721
- Print publication:
- December 2019
-
- Article
- Export citation
-
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 
${\text{gVP}}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Pi } _1 )$ + “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.
