Developments in constructive nonstandard analysis

Bulletin of Symbolic Logic 4 (3):233-272 (1998)
  Copy   BIBTEX

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

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,953

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

A Constructive View on Ergodic Theorems.Bas Spitters - 2006 - Journal of Symbolic Logic 71 (2):611 - 623.
Nonstandard set theory.Peter Fletcher - 1989 - Journal of Symbolic Logic 54 (3):1000-1008.
Second-order non-nonstandard analysis.J. M. Henle - 2003 - Studia Logica 74 (3):399 - 426.
Did Bishop have a philosophy of mathematics?Helen Billinge - 2003 - Philosophia Mathematica 11 (2):176-194.
Nonstandard combinatorics.Joram Hirshfeld - 1988 - Studia Logica 47 (3):221 - 232.
Nonstandard analysis and constructivism?Frank Wattenberg - 1988 - Studia Logica 47 (3):303 - 309.

Analytics

Added to PP
2009-01-28

Downloads
257 (#81,695)

6 months
11 (#272,636)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Infinitesimal Probabilities.Sylvia Wenmackers - 2016 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 199-265.
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.
Forcing in proof theory.Jeremy Avigad - 2004 - Bulletin of Symbolic Logic 10 (3):305-333.
Reverse formalism 16.Sam Sanders - 2020 - Synthese 197 (2):497-544.

View all 15 citations / Add more citations

References found in this work

Foundations of Constructive Analysis.John Myhill - 1972 - Journal of Symbolic Logic 37 (4):744-747.
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.

View all 6 references / Add more references