On the number of variables in undecidable superintuitionistic propositional calculi

TL;DR

This paper proves the existence of a minimal undecidable superintuitionistic propositional calculus with only three variables, filling a gap in the understanding of variable complexity in logical calculi.

Contribution

It constructs the first known undecidable superintuitionistic calculus with exactly three variables, establishing the minimal variable count for such undecidability.

Findings

Constructed a 3-variable undecidable superintuitionistic calculus

Proved no 2-variable superintuitionistic calculi exist

Established the minimal variable count for undecidability

Abstract

In this paper, we construct an undecidable 3-variable superintuitionistic propositional calculus, i.e., a finitely axiomatizable extension of the intuitionistic propositional calculus with axioms containing only 3 variables. Since there are no 2-variable superintuitionistic propositional calculi, this is the minimal possible number of variables.