Generic Vop\v{e}nka cardinals and models of ZF with few $\aleph_1$-Suslin sets

Abstract

We define a generic Vop\v{e}nka cardinal to be an inaccessible cardinal $\kappa$ such that for every first-order language $\mathcal{L}$ of cardinality less than $\kappa$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if $|\mathscr{B}| = \kappa$ and every structure in $\mathscr{B}$ has cardinality less than $\kappa$, then an elementary embedding between two structures in $\mathscr{B}$ exists in some generic extension of $V$. We investigate connections between generic Vop\v{e}nka 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 Vop\v{e}nka cardinal) is equiconsistent with ZF + $(2^{\aleph_1} \not\leq |S_{\aleph_1}|)$ where $S_{\aleph_1}$ is the pointclass of all $\aleph_1$-Suslin sets of reals, and also with ZF + $(S_{\aleph_1} = {\bf\Sigma}^1_2)$ + $(\Theta = \aleph_2)$ where $\Theta$ is the least ordinal that is not a surjective image of the reals.