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Foundational Aspects of "Non"standard Mathematics
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Contemporary Mathematics
1994; 135 pp; softcover
Volume: 176
ISBN-10: 0-8218-0293-3
ISBN-13: 978-0-8218-0293-9
List Price: US$48 Member Price: US$38.40
Order Code: CONM/176

This work proposes a major new extension of "non"standard mathematics. Addressed to a general mathematical audience, the book is intended to be philosophically provocative. The model theory on which "non"standard mathematics has been based is first reformulated within point set topology, which facilitates proofs and adds perspective. These topological techniques are then used to give new, uniform conservativity proofs for the various versions of "non"standard mathematics proposed by Nelson, Hrbáček, and Kawai. The proofs allow for sharp comparison. Addressing broader issues, Ballard then argues that what is novel in these forms of "non"standard mathematics is the introduction, however tentative, of relativity in one's mathematical environment. This hints at the possibility of a mathematical environment which is radically relativistic. The work's major and final feature is to present and prove conservative a version of "non"standard mathematics which, for the first time, illustrates this full radical relativism. The book is entirely self-contained, with all necessary background in point set topology, model theory, "non"standard analysis, and set theory provided in full.

General mathematical audiences.

Reviews

"The author contributes a number of sweeping ideas toward a new set-theoretic foundation for nonstandard analysis ... places a number of important topics in a new light."

-- Mathematical Reviews

"In this interesting research monograph the author shows that the theory of non-standard mathematics (Robinson, Nelson, etc.) can be developed from a topological viewpoint, and that, by this approach, new insight is gained. By adding ample background material and comments, the author has tried to make his exposition as accessible to a broad audience of mathematicians as possible."

-- Monatshefte für Mathematik

• Introduction
Part 1. Preliminaries
• Point set topology
• Model theory
• "Non"standard analysis
Part 2. Topological aspects
• Introduction
• Theory of CL spaces
• Topological determinacy of local internal domains
• Topological determinacy of internal domains
Part 3. Set theoretic aspects
• Introduction
• Standard set theory
• Current "non"standard set theories
• Proofs of conservativity
• Critical review with proposal: EST
• Conservativity of EST
• Concluding remarks
• References
• Index
• Symbols