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An axiomatics for nonstandard set theory, based on von Neumann–Bernays–Gödel Theory

Published online by Cambridge University Press:  12 March 2014

P. V. Andreev  and
E. I. Gordon
P. V. Andreev
Affiliation:
Xanadu Studio LLC, 4 Zubovski Blvd. Moscow, 119021, Russia, Phone: 7(095) 201-8972, E-mail: petr@xanadu.ru
E. I. Gordon
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA, Phone: 1(217) 333-9661, E-mail: gordon3@math.uiuc.edu
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Abstract

We present an axiomatic framework for nonstandard analysis—the Nonstandard Class Theory (NCT) which extends von Neumann–Gödel–Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms—related to it—analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1321 - 1341
Copyright
Copyright © Association for Symbolic Logic 2001

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