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

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 Vopěnka's Principle, are characterized by compactness properties involving Henkin models for sort logic.

This provides a model-theoretic analogy between Vopěnka's Principle and weak Vopěnka'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 Härtig quantifier logic $\mathbb L(I)$ can consistently be larger than the first supercompact cardinal.

Finally, we show that the upward Löwenheim-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.