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 ${\mathscr {B}}$ 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 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 + $(2^{\aleph _1} \not \le |S_{\aleph _1}|)$ where $S_{\aleph _1}$ is the pointclass of all $\aleph _1$-Suslin sets of reals, and also with ZF + $(S_{\aleph _1} = {{\varvec{\Sigma }}}^1_2)$ + $(\varTheta = \aleph _2)$ where $\varTheta$ is the least ordinal that is not a surjective image of the reals.