The Decision Problem for a Three-sorted Fragment of Set Theory with Restricted Quantification and Finite Enumerations

We solve the satisfiability problem for a three-sorted fragment of set theory (denoted ), which admits a restricted form of quantification over individual and set variables and the finite enumeration operator {-,-,...,-} over individual variables, by showing that it enjoys a small model property, i.e., any satisfiable formula ψ of has a finite model whose size depends solely on the length of ψ itself. Several set-theoretic constructs are expressible by -formulae, such as some variants of the power set operator and the unordered Cartesian product. In particular, concerning the latter construct, we show that when finite enumerations are allowed, the resulting formula is exponentially shorter than in their absence.