Model theory of class-sized logics

TL;DR

This paper explores the model-theoretic properties of class-sized logics, linking them to large cardinal axioms and showing how restrictions can recover classical properties.

Contribution

It characterizes large cardinals through model-theoretic properties of restricted class-sized logics, connecting set theory and logic.

Findings

Model-theoretic properties characterize large cardinals.

Some properties can be derived in ZFC without extra assumptions.

Restrictions on logic fragments restore classical model-theoretic properties.

Abstract

We study compactness and L\"owenheim-Skolem properties of fragments of the class-sized logic $\mathcal{L}_{\infty \infty}$ and of class-sized versions of second-order and sort logics. In these fragments, certain combinations of infinitary quantifiers and boolean connectives are banned. While model-theoretic properties fail for unrestricted class logics, this drastically changes in our more restricted setting. We show that model-theoretic properties of class logics characterise a wide array of large cardinals, and that some of them can even be obtained in ZFC. In particular, we give a characterisation of Weak Vop\v{e}nka's Principle and Ord is Woodin by downwards L\"owenheim-Skolem properties, and a characterisation of Shelah cardinals by a compactness property of class-sized logics. We further strengthen many known results about properties of set-sized logics by studying how they transfer to class-sized extensions.