Model-theoretic Characterizations of Large Cardinals

TL;DR

This paper explores how large cardinals can be characterized through compactness principles related to omitting types in various logical systems, extending known results to include huge cardinals with very high consistency strength.

Contribution

It provides new model-theoretic characterizations of large cardinals, including huge cardinals, via compactness for omitting types in advanced logics such as $L_{ abla, abla}$ and second-order logic.

Findings

Characterization of large cardinals using compactness for omitting types.

Extension of compactness characterizations to huge cardinals.

Analysis of second-order and sort logic in the context of large cardinal properties.

Abstract

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{\kappa, \kappa}$, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vop\v{e}nka's Principle.