The Cardinalities of Intervals of Equational Theories and Logics

TL;DR

This paper applies descriptive set theory to determine the cardinalities of classes of equational theories and logics, resolving several open problems and classifying their complexity within the Borel hierarchy.

Contribution

It establishes the Borel complexity of various classes of theories and logics, proving their cardinalities are either countable or continuum, and provides solutions to open problems in the field.

Findings

Intervals of equational theories are $oldsymbol{ ext{Pi}}^0_1$ sets.

Finite model property spans are $oldsymbol{ ext{Pi}}^0_2$ sets.

Set of pretabular extensions is a $oldsymbol{ ext{Pi}}^0_2$ set.

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 [Trans. Am. Math. Soc. 370 (2018), pp. 4785-4812] regarding the existence of independent systems, which relies on Je\v{z}ek et al. [Bull. Aust. Math. Soc. 42 (1990), pp. 57-70].