Model-theoretic characterizations of large cardinals (Re) ²visited

We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that ₙ-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vopenka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vopenka's Principle and weak Vopenka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic L (Q^WF), and show that the compactness number of the H\"artig quantifier logic 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 L² and the sort logic L^s, n are given by the first extendible and C^ (n) -extendible cardinal, respectively.