Reflection with Absolute Generality
Abstract: Traditionally, reflection principles in set theory claim that the set-theoretic universe is indescribable. It is natural to consider reflection principles with absolute generality, which asserts that the universe containing everything, including sets and urelements, is indescribable. In the first part of this talk, I will consider the first-order reflection principle in urelement set theory. With the Axiom of Choice, first-order reflection holds just in case the sets of urelements are arranged in a certain way, and this equivalence falls apart without AC. In the second part of this talk, I will present my joint work with Joel Hamkins on second-order reflection principles with urelements. A standard version of second-order reflection, due to Paul Bernays, is often considered as a weak large cardinal axiom in set theory. With abundant urelements, however, Bernays’ second-order reflection principle bi-interprets a supercompact cardinal.
The talk will be hybrid in Oslo and on Zoom.
Contact Laura.Crosilla @ ifikk.uio.no for a Zoom-invite.