Bulletin of Symbolic Logic

Developments in Constructive Nonstandard Analysis

Erik Palmgren

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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.

Article information

Source
Bull. Symbolic Logic, Volume 4, Number 3 (1998), 233-272.

Dates
First available in Project Euclid: 20 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1182353577

Mathematical Reviews number (MathSciNet)
MR1650475

Zentralblatt MATH identifier
0920.03063

JSTOR
links.jstor.org

Citation

Palmgren, Erik. Developments in Constructive Nonstandard Analysis. Bull. Symbolic Logic 4 (1998), no. 3, 233--272. https://projecteuclid.org/euclid.bsl/1182353577


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