A characterization of internal sets (Q1262312)

The paper deals with a universe of sets satisfying the usual axioms of Zermelo-Fraenkel set theory with atoms (urelements). It is assumed that \({\mathbb{R}}\), the set of real numbers, is represented by a set of atoms. If in this universe of sets, A is the superstructure based on \({\mathbb{R}}\), then the theory \({\mathcal A}:=(A;\in | A^ 2;id| A^ 2)\) is called ``standard analysis''. The author considers nonstandard models of the theory of \({\mathcal A}\) that are proper extensions of \({\mathcal A}\). The main result of the paper is a characterization of the class of internal sets as the unique subclass of the model that contains the standard sets, is transitive and membership well-founded.