Developments in constructive nonstandard analysis

Bulletin of Symbolic Logic 4 (3):233-272 (1998)
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
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DOI 10.2307/421031
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References found in this work BETA
A Model for Intuitionistic Non-Standard Arithmetic.Ieke Moerdijk - 1995 - Annals of Pure and Applied Logic 73 (1):37-51.
A Constructive Approach to Nonstandard Analysis.Erik Palmgren - 1995 - Annals of Pure and Applied Logic 73 (3):297-325.
Nonstandard Analysis and Constructivism?Frank Wattenberg - 1988 - Studia Logica 47 (3):303 - 309.
A Sheaf-Theoretic Foundation for Nonstandard Analysis.Erik Palmgren - 1997 - Annals of Pure and Applied Logic 85 (1):69-86.

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Citations of this work BETA
A Functional Interpretation for Nonstandard Arithmetic.Benno van den Berg, Eyvind Briseid & Pavol Safarik - 2012 - Annals of Pure and Applied Logic 163 (12):1962-1994.
Saturated Models of Intuitionistic Theories.Carsten Butz - 2004 - Annals of Pure and Applied Logic 129 (1-3):245-275.

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