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 \u3c6{symbol}, \u3b5, =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of \u3c6{symbol}, \u3b5, =, {...}, \u3b7, where {...} is the tuple operator and \u3b7 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 \u3c6{symbol}, \u3b5, =, {...}, \u3b7, obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving \u3b7 is allowed