Modal Logics That Need Very Large Frames

Abstract

The Kuznetsov-Index of a modal logic is the least cardinal $\mu$ such that any consistent formula has a Kripke-model of size $\leq \mu$ if it has a Kripke-model at all. The Kuznetsov-Spectrum is the set of all Kuznetsov-Indices of modal logics with countably many operators. It has been shown by Thomason that there are tense logics with Kuznetsov-Index $\beth_{\omega+\omega}$. Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index $\beth_{\omega}$. We will show here that for each countable ordinal $\lambda$ there are logics with Kuznetsov-Index $\beth_{\lambda}$. Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for $\Pi^1_1$-theories which is likewise defined. A particular consequence is the following. If inaccessible (weakly compact, measurable) cardinals exist, then the least inaccessible (weakly compact, measurable) cardinal is also a Kuznetsov-Index.