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

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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, Németi, 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, Gödels’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).

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Notes

  1. Vaught’s theorem, an easy consequence of the Henkin–Orey \(\textsf {OTT}\) says that countable atomic theories have countable models.

  2. However, some logicians do not agree to this sweeping statement. Quoting Shelah on this: People say that settling Vaught’s conjecture is the most important problem in Model theory, because it makes us understand countable models of countable theories, which are the most important models. We disagree with all three statements.

  3. A theory is defined to be totally transcendental \(\iff \) every formula has a Morley rank; furthermore, one can prove that in a stable totally transcendental theory each formula has finite Morely rank. If \(\mathcal L\) is a countable first order language, then totally transcendental theories are precisely the \(\aleph _0\) stable ones. However, if \(\mathcal L\) is uncountable, then for an \(\mathcal L\) theory, “being totally transcaendtal ” is not equivalent with “being \(\aleph _0\) stable”. In other words, these two notions are in general distinct. In fact a totally transcendental theory T is stable, if T is stable for every \(\lambda \ge |T|\). For related definitions and further results, cf. [56, Theorem II.3.1 and Conclusion II.3.3 (2) on p. 41–42].

  4. An analogous proof is obtained independently by Gabor Sági.

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  1. Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt

    Tarek Sayed Ahmed

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Ahmed, T.S. A brief history of Tarskian algebraic logic with new perspectives and innovations. Boll Unione Mat Ital 13, 381–416 (2020). https://doi.org/10.1007/s40574-020-00240-x

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