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On the strength of nonstandard analysis

Published online by Cambridge University Press:  12 March 2014

C. Ward Henson  and
H. Jerome Keisler
C. Ward Henson
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
H. Jerome Keisler
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
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Extract

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.

The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 377 - 386
Copyright
Copyright © Association for Symbolic Logic 1986

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References

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