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Standard foundations for nonstandard analysis

Published online by Cambridge University Press:  12 March 2014

David Ballard  and
Karel Hrbacek
David Ballard
Affiliation:
Department of Mathematics, Sonoma State University, Rohnert Park, California 94926
Karel Hrbacek
Affiliation:
Department of Mathematics, City College, Cuny, New York, New York 10031
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Extract

In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 741 - 748
Copyright
Copyright © Association for Symbolic Logic 1992

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References

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