A universal approach to Omitting types for various multimodal and quantifier logics

TL;DR

This paper develops a universal algebraic framework to study the omitting types theorem across various multimodal and quantifier logics, revealing both positive and negative results and connections to set-theoretic axioms.

Contribution

It introduces a universal algebraic approach to analyze the omitting types property in diverse logical systems, extending classical results and exploring their set-theoretic independence.

Findings

Countable theories with quantifier elimination can omit fewer than continuum many types.

Eliminating maximality conditions relates to independence from ZFC, implying set-theoretic considerations.

Some algebraic reformulations of the omitting types theorem hold, others fail, depending on the logic variant.

Abstract

We intend to investigate the metalogical property of 'omitting types' for a wide variety of quantifier logics (that can also be seen as multimodal logics upon identifying existential quantifiers with modalities syntactically and semantically) exhibiting the essence of its abstract algebraic facet, namely, atom-canonicity, the last reflecting a well known persistence propery in modal logic. In the spirit of 'universal logic ', with this algebraic abstraction at hand, the omitting types theorem OTT will be studied for various reducts extensions and variants (possibly allowing formulas of infinite length) of first order logic. Our investigatons are algebraic, addressing (non) atom canoicity of varieties of algebra of relations. In the course of our investigations, both negative and positive results will be presented. For example, we show that for any countable theory $L_n$ theory $T$ that has quantifier elimination $< 2^{\omega}$ many non-principal complete types can be omitted. Furthermore, the maximality (completeness) condition, if eliminated, leads to an independent statement from $\sf ZFC$ implied by Martin's axiom. $\sf OTT$s are approached for other algebraizable (in the classical Blok-Pigozzi sense) reformulations or/ and versions of $L_{\omega \omega}$; they are shown to hold for some and fail for others. We use basic graph theory, finite combinatorics, combinatorial game theory orchestrated by algebraic logic.