Nonstandard Analysis, AxiomaticallyIn 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. |
Contents
Table of Contents | 1 |
Getting started | 11 |
3 | 54 |
2 | 73 |
Historical and other notes to Chapter 2 | 81 |
1f Remarks on Basic Idealization and Saturation | 92 |
3 | 105 |
5 | 118 |
4 | 166 |
Definable external sets and metamathematics of | 179 |
Partially saturated universes and the Power Set problem | 219 |
Forcing extensions of the nonstandard universe | 257 |
Other nonstandard theories | 289 |
Hyperfinite descriptive set theory | 317 |
| 389 | |
| 392 | |
Other editions - View all
Nonstandard Analysis, Axiomatically Vladimir Grigorʹevich Kanoveĭ,Michael Reeken No preview available - 2004 |
Common terms and phrases
A-codes argument assume belongs bijection Borel sets cardinal cofinal Collection considered contains Corollary countably determined sets cross-sections defined definition descriptive set theory domain E-formula EEST elementarily equivalent elementary elements equivalence relation Exercise exists extension of ZFC extensional external sets finite set formula hence Hrbaček hyperfinite implies induction infinite internal sets Inner Transfer internal core extension internal subsets internal subuniverse isomorphic KanR Lemma Loeb measure natural numbers nonstandard analysis nonstandard set theories Note ordinal parameters Polish descriptive set Polish spaces proof of Theorem proper class properties prove quotient resp result satisfying Saturation sequence set of standard set universe st-E-definable st-E-formula standard core extension standard core interpretation standard set structure subsets of H Suppose Theorem transitive transitive set ultrafilter ultrapower well-founded sets ZFBC ZFC universe
References to this book
Mathematical Logic in Asia: Proceedings of the 9th Asian Logic Conference ... S. S. Goncharov,Rod G. Downey,H. Ono No preview available - 2006 |