Decidability of the Satisfiability Problem for Boolean Set Theory with the Unordered Cartesian Product Operator

Abstract

We give a positive solution to the decidability problem for the fragment of set theory, dubbed BST ⊗, consisting of quantifier-free formulae involving the Boolean set operators of union, intersection, and set difference, along with the unordered Cartesian product operator ⊗ (where \(s \otimes t := \big \lbrace \lbrace u,v\rbrace \,\texttt {|}\:u \in s \wedge v \in t \big \rbrace\) ), and the equality predicate, but no membership. Specifically, we provide nondeterministic exponential decision procedures for both the ordinary and the finite satisfiability problems for BST ⊗. We expect that these decision procedures can be adapted for the standard Cartesian product and, with added technicalities, to the cases involving membership, providing a solution to a longstanding problem in computable set theory.