Nonstandard Analysis, Axiomatically

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.

Includes supplementary material: sn.pub/extras

Auteur

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 Universität Wuppertal since 1979

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

Texte du rabat

The book is devoted to nonstandard set theories that serve as foundational basis for nonstandard mathematics. Several popular and some less known nonstandard theories are considered, including internal set theory IST, Hrbacek set theory HST, and others. The book presents the basic structure of the set universe of these theories and methods to effectively develop "applied" nonstandard analysis, metamathematical properties and interrelations of these nonstandard theories between each other and with ZFC and some variants of ZFC, foundational problems of the theories, including the problem of external sets and the Power Set problem, and methods of their solution. The book is oriented towards a reader having some experience in foundations (set theory, model theory) and in nonstandard analysis.

Résumé
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.

Contenu
1 Getting started.- 2 Elementary real analysis in the nonstandard universe.- 3 Theories of internal sets.- 4 Metamathematics of internal theories.- 5 Definable external sets and metamathematics of HST.- 6 Partially saturated universes and the Power Set problem.- 7 Forcing extensions of the nonstandard universe.- 8 Other nonstandard theories.- 9 Hyperfinite descriptive set theory.- References.