Decision procedures for elementary sublanguages of set theory: XIII. Model graphs,reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on , , =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of , , =, {...}, , where {...} is the tuple operator and is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of , , =, {...}, , obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving is allowed.