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