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Developments in Constructive Nonstandard Analysis

Published online by Cambridge University Press:  15 January 2014

Erik Palmgren
Erik Palmgren*
Affiliation:
Department of Mathematics, Uppsala University, P.O. BOX 480, S-751 06 Uppsala, Sweden.E-mail: [email protected]
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Abstract

We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine–Borel theorem, the Cauchy–Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 4 , Issue 3 , September 1998 , pp. 233 - 272
Copyright
Copyright © Association for Symbolic Logic 1998

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