Getting started
This chapter introduces HST, Hrbaček set theory.
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Vladimir Kanovei • Michael Reeken
Nonstandard Analysis, Axiomatically
~ Springer
Vladimir Kanovei
IITP, Institute for Information Transmission Bol. Karetny 19, 127994 Moscow Russian Federation Michael Reeken
Bergische Universităt Wuppertal FB C Mathematik GauBstr. 20,42119 Wuppertal Germany
Library of Congress Control Number: 2004109574 Mathematics Subject Classification (2000): Primary: 03E, 03C, 03H05, 03E70 Secondary: ooA30, ooA35. 26E35. 28E05. 54J05 ISBN 978-3-642-06077-9
ISBN 978-3-662-08998-9 (eBook)
DOI 10.1007/978-3-662-08998-9
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© Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover Ist edition 2004
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Preface
In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at standard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathematical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation. One of the crucial discoveries in foundations was that the structures studied in mathematics do have nonstandard models. Starting with A. Rob
