2025/05/21 by Boney, Will, Osinski, Jonathan
#03C85 #03C95 #03E55 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2505.15574
We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that Πn-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(QWF), 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 L2 and the sort logic \mathbb Ls,n are given by the first extendible and C(n)-extendible cardinal, respectively.