Compactness characterisations of large cardinals with strong Henkin models

Abstract

We consider compactness properties for strong logics in terms of <em>strong Henkin models</em> and give characterisations of supercompact cardinals, $\mathrm{C}^{(n)}$-extendible cardinals, and Vopěnka's Principle by these properties. Moreover, we give a characterisation of superstrong cardinals in terms of compactness properties using the previously considered <em>weak Henkin models</em>.