Nonstandard Analysis, Axiomatically

1. Front Matter

   Pages I-XVI

   PDF

2. Introduction

   • Vladimir Kanovei, Michael Reeken

   Pages 1-10

3. Getting started

   • Vladimir Kanovei, Michael Reeken

   Pages 11-52

4. Elementary real analysis in the nonstandard universe

   • Vladimir Kanovei, Michael Reeken

   Pages 53-82

5. Theories of internal sets

   • Vladimir Kanovei, Michael Reeken

   Pages 83-129

6. Metamathematics of internal theories

   • Vladimir Kanovei, Michael Reeken

   Pages 131-178

7. Definable external sets and metamathematics of HST

   • Vladimir Kanovei, Michael Reeken

   Pages 179-217

8. Partially saturated universes and the Power Set problem

   • Vladimir Kanovei, Michael Reeken

   Pages 219-255

9. Forcing extensions of the nonstandard universe

   • Vladimir Kanovei, Michael Reeken

   Pages 257-287

10. Other nonstandard theories

    • Vladimir Kanovei, Michael Reeken

    Pages 289-315

11. “Hyperfinite” descriptive set theory

    • Vladimir Kanovei, Michael Reeken

    Pages 317-388

12. Back Matter

    Pages 389-410

    PDF

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.

• Nonstandard
• ZFC
• calculus
• differential equation
• forcing
• foundations
• model theory
• problems
• set theory

• IITP, Institute for Information Transmission, Moscow, Russian Federation

  Vladimir Kanovei

• Bergische Universität Wuppertal, FB C Mathematik, Wuppertal, Germany

  Michael Reeken

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.

 

 

Policies and ethics