The Cardinalities of Intervals of Equational Theories and Logics

Abstract

We study the cardinality of classes of equational theories (varieties) and logics by applying descriptive set theory. We affirmatively solve open problems raised by Jackson and Lee [Trans. Am. Math. Soc. 370 (2018), pp. 4785-4812] regarding the cardinalities of subvariety lattices, and by Bezhanishvili et al. [J. Math. Log. (2025), in press] regarding the degrees of the finite model property (fmp). By coding equations and formulas by natural numbers, and theories and logics by real numbers, we examine their position in the Borel hierarchy. We prove that every interval of equational theories in a countable language corresponds to a $\boldsymbol{\Pi}^0_1$ set, and every fmp span of a normal modal logic to a $\boldsymbol{\Pi}^0_2$ set. It follows that they have cardinality either $\leq \aleph_0$ or $2^{\aleph_0}$, provably in ZFC. In the same manner, we observe that the set of pretabular extensions of a tense logic is a $\boldsymbol{\Pi}^0_2$ set, so its cardinality is either $\leq \aleph_0$ or $2^{\aleph_0}$. We also point out a negative solution to another open problem raised by Jackson and Lee, op. cit., regarding the existence of independent systems, which relies on Je\v{z}ek et al. [Bull. Aust. Math. Soc. 42 (1990), pp. 57-70].