Modal Logics that Need Very Large Frames
BibTeX
@MISC{Kracht_modallogics,
author = {Marcus Kracht},
title = {Modal Logics that Need Very Large Frames},
year = {}
}
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Abstract
. The Kuznetsov{Index of a modal logic is the least cardinal such that any consistent formula has a Kripke{model of size 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 i!+! . Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov{Index i! . We will show here that for each countable ordinal there are logics with Kuznetsov{Index i . Furthermore, we show that the Kuznetsov{Spectrum is identical to the spectrum of indices for 1 1 {theories, which is likewise dened. 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. 1. Introduction Suppose ' is an elementary formula and that ' is consistent with an elementary theory T in a countable language. Then t...
