## Logic Seminar

Trevor Wilson

Miami University

Generic Vopenka cardinals and models with few Suslin sets

**Abstract:**A generic Vopenka cardinal is an inaccessible cardinal kappa such that for every kappa-sequence of structures in V_kappa in the same first-order language, an elementary embedding between two of the structures exists in some generic extension of V. Because the elementary embedding is not required to exist in V, this is a rather weak large cardinal property: if $0^\sharp$ exists, then every Silver indiscernible is a generic Vopenka cardinal in L. We show that generic Vopenka cardinals are closely related to a matter in descriptive set theory, namely the number of Suslin sets of reals in models of ZF without the axiom of choice. In particular, we show that ZFC + "there is a generic Vopenka cardinal" is equiconsistent with ZF + DC + "there is no injection from $P(\omega_1)$ to the pointclass of Suslin sets."

Tuesday February 13, 2018 at 3:30 PM in SEO 427