Generic Vopěnka cardinals and models of ZF with few ${\aleph }_1$-Suslin sets

Abstract

We define a generic Vopěnka cardinal to be an inaccessible cardinal $\kappa$ such that for every first-order language $\mathcal{L}$ of cardinality less than $\kappa$ and every set $\mathcal{B}$ of $\mathcal{L}$-structures, if $|\mathcal{B}|=\kappa$ and every structure in $\mathcal{B}$ has cardinality less than $\kappa$, then an elementary embedding between two structures in $\mathcal{B}$ exists in some generic extension of $V$. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of ${\aleph }_1$-Suslin sets of reals in models of ZF. In particular, we show that ZFC + (there is a generic Vopěnka cardinal) is equiconsistent with ZF + $(2^{{\aleph }_1}\nleqq |S_{{\aleph }_1}|)$ where $S_{{\aleph }_1}$ is the pointclass of all ${\aleph }_1$-Suslin sets of reals, and also with ZF + $(S_{{\aleph }_1}={\mathbf{\Sigma }}_2^1)$ + $(\Theta ={\aleph }_2)$ where $\Theta$ is the least ordinal that is not a surjective image of the reals.