Nonstandard Analysis, Axiomatically

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Springer Science & Business Media, Aug 11, 2004 - Mathematics - 408 pages
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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 stan dard 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 mathemat ical 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.
  

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Contents

Table of Contents
1
Getting started
11
Elementary real analysis in the nonstandard universe
53
11
79
Theories of internal sets
83
Metamathematics of internal theories
131
22
162
Definable external sets and metamathematics of
179
Other nonstandard theories
289
Hyperflnite descriptive set theory
317
28
321
29
334
42
343
References
389
Index
397
47
398

22
181
24
208
Partially saturated universes and the Power Set problem
219

Common terms and phrases

References to this book

About the author (2004)

I) Vladimir Kanovei

graduated Moscow State university 1973
PhD Moscow State university 1976
Doctor of Science in Phys. Math. Moscow Steklov inst. 1986

assistant to full professor at Moscow Railroad engineering inst.
1976 - 1998

currently leading researcher at Institute for Information
transmissin problems (IITP) Moscow

interests in mathematics:
logic and foundations, set theory, nonstandard analysis
publications:
over 100 papers in Russian and international mathematical journals

II)
Michael Reeken
PhD in theoretical physics, University of Vienna 1968

Research Fellow at the Battelle Institute, Geneva, 1969 - 1972

Research grant at the University of Bonn, 1972 - 1974

Professor at the University of Bochum, 1972 - 1979

Full Professor at the Bergische Universitšt Wuppertal since 1979

interests in mathematics:
problems from mathematical physics, nonlinear functional analysis,
nonstandard mathematics, philosophy of mathematics.

Bibliographic information