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A characterization of internal sets. (English) Zbl 0686.03039

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.
Reviewer: W.A.J.Luxemburg

MSC:

03H05 Nonstandard models in mathematics
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References:

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