Characterizing large cardinals in terms of Löwenheim-Skolem and weak compactness properties of strong logics

Trevor Wilson (Miami U)

Jun 26. 2024, 09:15 — 10:00

We present characterizations of large cardinals in terms of properties of strong logics, such as Löwenheim-Skolem and weak compactness properties of certain fragments of infinitary second-order logic.  For example, a cardinal kappa is strong if and only if the Löwenheim-Skolem property holds at kappa for a certain fragment of infinitary second-order logic, namely for the sentences whose negation normal form has only <kappa-length conjunctions and finitary existential quantifiers (but arbitrary-length disjunctions and universal quantifiers.)

We also present recent work with Jonathan Osinski obtaining a related result for a kind of weak compactness property.  Namely, a cardinal kappa is Shelah if and only if the following property holds: for every theory of size kappa in the dual of the aforementioned logic fragment, if every subtheory of size less than kappa has a model of size less than kappa, then the theory itself has a model.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Determinacy, Inner Models and Forcing Axioms (Workshop)
Organizer(s):
Sandra Müller (TU Vienna)
Grigor Sargsyan (Polish Academy of Science, Warsaw)
Ralf Schindler (WWU Münster)
John Steel (UC, Berkeley)