Modal Logics That Need Very Large Frames

Notre Dame Journal of Formal Logic 40 (2):141-173 (1999)
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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 . Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index . We will show here that for each countable ordinal there are logics with Kuznetsov-Index . Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for -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

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10.1305/ndjfl/1038949533

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Citations of this work

Technical Modal Logic.Marcus Kracht - 2011 - Philosophy Compass 6 (5):350-359.

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[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
The modal logic of inequality.Maarten de Rijke - 1992 - Journal of Symbolic Logic 57 (2):566-584.
Properties of independently axiomatizable bimodal logics.Marcus Kracht & Frank Wolter - 1991 - Journal of Symbolic Logic 56 (4):1469-1485.
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