Modal logics that need very large frames

Abstract

The Kuznetsov-Index of a modal logic is the least cardinal µ such that any consistent formula has a Kripke-model of size ≤ µ 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 ℶω+ω. Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index ℶω. We will show here that for each countable ordinal λ there are logics with Kuznetsov-Index ℶλ. Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for (formula)-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. © 1999 by the University of Notre Dame. All rights reserved.