A brief history of Tarskian algebraic logic with new perspectives and innovations

Abstract

We take a magical tour in algebraic logic, which is the natural interface between universal algebra and mathematical logic, starting from classical results on neat embeddings due to Andreḱa, Henkin, Nemeti, Monk and Tarski, all the way to recent novel results in algebraic logic using so-called rainbow constructions. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics dealt with include the interaction of algebraic and modal logic, the so-called (central still active) finitizability problem, Godels’s incompleteness Theorem in guarded fragments, counting the number of subvarieties of $$\textsf {RCA}_{\omega }$$ which is reminiscent of Shelah’s classification theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught’s conjecture in model theory. The paper has a survey character but it contains new results and new approaches to old ones (such as the interaction of algebraic logic and descriptive set theory).