Vop\v{e}nka's Principle without Choice: Preservation under Symmetric Extensions

Abstract

We prove that every set-sized symmetric extension of a model of $\mathsf{ZF}+\mathsf{VP}$ again satisfies $\mathsf{ZF}+\mathsf{VP}$, where $\mathsf{VP}$ is formulated for arbitrary set-sized languages. For each standard $n\geq 1$, the same preservation theorem holds for $\mathsf{VP}(\Pi_n)$, with set parameters and arbitrary set-sized languages. We also prove that $\mathsf{ZF}+\mathsf{VP}$ and $\mathsf{ZFC}+\mathsf{VP}$ are equiconsistent, and that $\mathsf{ZF}+\mathsf{VP}$ proves the existence of a proper class of L\"owenheim-Skolem cardinals. Relative to $\operatorname{Con}(\mathsf{ZF}+\mathsf{VP})$, the axiom $\mathsf{DC}$ is independent of $\mathsf{ZF}+\mathsf{VP}+\neg\mathsf{AC}$. The preservation theorem also yields Feferman-L\'evy and full Solovay models satisfying $\mathsf{VP}$. It remains open whether every countable model of $\mathsf{ZF}+\mathsf{VP}$ has a class-generic extension satisfying $\mathsf{ZFC}+\mathsf{VP}$.