Upcoming Events

Logic Colloquium

May 02, 2025, 4:10 PM

60 Evans Hall

Omer Ben-Neria
UCLA

On the strength of ultrafilters on ordinals above choiceless large cardinals

Kunen famously showed that the Axiom of Choice (AC) places fundamental limitations on the large cardinal hierarchy, proving that ZFC is incompatible with the existence of an elementary embedding from V to itself (a property defining Reinhardt cardinals). However, even without the axiom of choice, recent developments have uncovered a rich structural theory for sets and cardinals above choiceless large cardinals. Goldberg established the existence of a proper class of regular cardinals and showed that each cardinal in a tail segment of this class carries a sigma-complete uniform ultrafilter. This naturally leads to questions concerning the strength of ultrafilters on ordinals in this choiceless setting. Within ZFC, the strength of a sigma-complete ultrafilter is closely tied to combinational characteristics linking it with large cardinal properties, such as measurability, strong compactness, and supercompactness. In this talk, I will present several results that limit the possible strength of ultrafilters on ordinals above choiceless large cardinals. Furthermore, I will discuss applications of these findings to the theory of Prikry forcings and their iteration in this context, and in particular, outline new constructions of classical consistency results in choiceless set theory, including a “Gitik model” in which all uncountable cardinals are singular. This is joint work with William Adkisson.

Logic Colloquium

May 09, 2025, 4:10 PM

60 Evans Hall

Johan van Benthem
Stanford University

Mathematical Logic Meets Philosophy: Tracing the role of Beth Definability and Craig Interpolation up to today

Philosophical logic is often seen as the sole guardian of the interface of logic and philosophy. But classical results from mathematical logic also play a major role. We illustrate this for the case of the Beth Definability Theorem and the Craig Interpolation Theorem, and how these have occurred since the 1950s in different areas, from philosophy of science to epistemology, metaphysics, and philosophy of language. Reflecting on this encounter, we point out new results and issues, including connections with situation theory, logics of (in-)dependence, and translations between formal systems.