Degrees of the finite model property: the antidichotomy theorem

2023/07/14 by Bezhanishvili, Guram, Bezhanishvili, Nick, Moraschini, Tommaso · 1 citation
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2307.07209

Abstract

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic \sf K is 1 or 2^ℵ₀. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as \sf S4 or \sf K4) or for extensions of the intuitionistic propositional calculus IPC. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of \sf K remains 1 or 2^ℵ₀. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of IPC: each nonzero cardinal κ such that κ≤ ℵ₀ or κ= 2^ℵ₀ is realized as the degree of fmp of some extension of IPC. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of \sf S4 and \sf K4.

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