March 8
Jonathan Osinski,
Univeristy of Hamburg
Model Theory of class-sized logics
We consider logics in which the collection of sentences over a set-sized vocabulary can form a proper class. The easiest example of such a logic is $\mathcal L_{\infty \infty}$, which allows for disjunctions and conjunctions over arbitrarily sized sets of formulas and quantification over strings of variables of any infinite length. Model theory of $\mathcal L_{\infty \infty}$ is very restricted. For instance, it is inconsistent for it to have nice compactness or Löwenheim-Skolem properties. However, Trevor Wilson recently showed that the existence of a Löwenheim-Skolem-Tarski number of a certain class-sized fragment of $\mathcal L_{\infty \infty}$ is equivalent to the existence of a supercompact cardinal, and various other related results. We continue this work by considering several appropriate class-sized logics and their relations to large cardinals. This is joint work with Trevor Wilson.