Model-theoretic characterizations of large cardinals (Re)${}^2$visited
Will Boney, Jonathan Osinski
TL;DR
The paper builds a precise bridge between large-cardinal hierarchies and model-theoretic properties of logics beyond first order. It shows that $\Pi_n$-strong cardinals are captured by Henkin-compactness for $\mathbb{L}^{s,n}$, and that weak Vopěnka's Principle aligns with this Henkin-compactness pattern, mirroring VP via a model-theoretic lens. It proves that huge cardinals arise from type-omission compactness in $\mathbb{L}(Q^{\text{WF}})$, and that the compactness number of $\mathbb{L}(I)$ can exceed the first supercompact under suitable forcing, while also equating the compactness of $\mathbb{L}(Q^{\text{WF}},I)$ with $\mathbb{L}(I)$. Finally, it identifies the upward Löwenheim-Skolem-Tarski numbers of $\mathbb{L}^2$ and $\mathbb{L}^{s,n}$ with the first extendible and the first $C^{(n)}$-extendible cardinals, respectively, providing a cohesive, localizable model-theoretic portrait of the VP/WVP landscape.
Abstract
We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $Π_n$-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.