Bulletin of Symbolic Logic

Developments in Constructive Nonstandard Analysis

Erik Palmgren

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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.

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

First available in Project Euclid: 20 June 2007

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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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