Formative processes with applications to the decision problem in set theory: II. Powerset and singleton operators, finiteness predicate

In this paper we solve the satisfiability problem for a quantifier-free fragment of set theory involving the powerset and the singleton operators and a finiteness predicate, besides the basic Boolean set operators of union, intersection, and difference. The more restricted fragment obtained by dropping the finiteness predicate has been shown to have a solvable satisfiability problem in a previous paper, by establishing for it a small model property. To deal with the finiteness predicate we have formulated and proved a small model witness property for our fragment of set theory, namely a property which asserts that any satisfiable formula of our fragment has a model admitting a “small” representation.