Model-theoretic characterizations of large cardinals (Re)${}^2$visited

TL;DR

This paper provides model-theoretic characterizations of various large cardinal notions, linking them to properties of extensions of first-order logic and related logical systems.

Contribution

It introduces new model-theoretic characterizations of large cardinals, including strong, huge, and extendible cardinals, via properties of Henkin models and compactness in extended logics.

Findings

Characterizes $ ext{Pi}_n$-strong cardinals through Henkin model compactness.

Links huge cardinals to compactness for type omission in well-foundedness logic.

Shows the L"owenheim-Skolem-Tarski number for second-order logic relates to the first extendible cardinal.

Abstract

We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $\Pi_n$-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vop\v{e}nka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vop\v{e}nka's Principle and weak Vop\v{e}nka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic $\mathbb L(Q^{\text{WF}})$, and show that the compactness number of the H\"artig quantifier logic $\mathbb L(I)$ can consistently be larger than the first supercompact cardinal. Finally, we show that the upward L\"owenheim-Skolem-Tarski number of second-order logic $\mathbb L^2$ and the sort logic $\mathbb L^{s,n}$ are given by the first extendible and $C^{(n)}$-extendible cardinal, respectively.