Nonstandard Analysis, Axiomatically
Springer Science & Business Media, Aug 11, 2004 - Mathematics - 408 pages
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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Table of Contents
Elementary real analysis in the nonstandard universe
Theories of internal sets
Metamathematics of internal theories
Deﬁnable external sets and metamathematics of
A-code argument axiom Basic ldealization belongs bijection Borel sets cardinal coﬁnal Collection consider contains Corollary countably determined sets deﬁnable deﬁned deﬁnition descriptive set theory domain domP E-formula EEST elementary embedding elements equiconsistent equivalence relation Exercise exists extension of ZFC external sets ﬁnd ﬁnite set ﬁrst formula hence hyperﬁnite implies induction inﬁnitesimal Inner Standardization Inner Transfer internal subsets internal subuniverse interpretation of BST isomorphic KanR Lemma Loeb measure metamathematical n-size natural numbers Nms(P nonstandard analysis notion ordinal parameters Polish spaces Power Set proof of Theorem proper class prove quantiﬁers quotient power resp result satisﬁes satisfying Saturation sequence set of standard set universe set X Q st-E-formula standard core extension standard core interpretation standard set structure sufﬁces Suppose Theorem transitive transitive set ultraﬁlter ultrapower well-founded sets ZFGT