Approaches to analysis with infinitesimals following Robinson, Nelson, and others

Peter Fletcher; Karel Hrbacek; Vladimir Kanovei; Mikhail G. Katz; Claude Lobry; Sam Sanders

doi:10.14321/realanalexch.41.1.0193

Approaches to analysis with infinitesimals following Robinson, Nelson, and others

Peter Fletcher
, Karel Hrbacek
, Vladimir Kanovei
, Mikhail G. Katz
, Claude Lobry
, Sam Sanders

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis.We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.

Original language	English
Pages (from-to)	193-252
Number of pages	60
Journal	Real Analysis Exchange
Volume	42
Issue number	2
DOIs	https://doi.org/10.14321/realanalexch.41.1.0193
State	Published - 2017

Keywords

Axiomatisations
Ideal elements
Infinitesimal
Intuitionism
Multiverse
Naive integers
Nonstandard analysis
Protozoa
Set-theoretic foundations
Soritical properties
Superstructure
Ultraproducts

ASJC Scopus subject areas

Analysis
Geometry and Topology

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10.14321/realanalexch.41.1.0193

Cite this

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title = "Approaches to analysis with infinitesimals following Robinson, Nelson, and others",

abstract = "This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis.We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.",

keywords = "Axiomatisations, Ideal elements, Infinitesimal, Intuitionism, Multiverse, Naive integers, Nonstandard analysis, Protozoa, Set-theoretic foundations, Soritical properties, Superstructure, Ultraproducts",

author = "Peter Fletcher and Karel Hrbacek and Vladimir Kanovei and Katz, \{Mikhail G.\} and Claude Lobry and Sam Sanders",

note = "Funding Information: We are grateful to Eric Leichtnam and Dalibor Praˇz{\'a}k for helpful comments on an earlier version of the manuscript. Vladimir Kanovei was supported in part by the RFBR grant number 17-01-00705. M. Katz was partially supported by Israel Science Foundation grant no. 1517/12. S. Sanders was supported by the following funding bodies: FWO Flanders, the John Templeton Foundation, the Alexander von Humboldt Foundation, and the Japan Society for the Promotion of Science, and expresses gratitude towards these institutions.",

year = "2017",

doi = "10.14321/realanalexch.41.1.0193",

language = "English",

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pages = "193--252",

journal = "Real Analysis Exchange",

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T1 - Approaches to analysis with infinitesimals following Robinson, Nelson, and others

AU - Fletcher, Peter

AU - Hrbacek, Karel

AU - Kanovei, Vladimir

AU - Katz, Mikhail G.

AU - Lobry, Claude

AU - Sanders, Sam

N1 - Funding Information: We are grateful to Eric Leichtnam and Dalibor Praˇzák for helpful comments on an earlier version of the manuscript. Vladimir Kanovei was supported in part by the RFBR grant number 17-01-00705. M. Katz was partially supported by Israel Science Foundation grant no. 1517/12. S. Sanders was supported by the following funding bodies: FWO Flanders, the John Templeton Foundation, the Alexander von Humboldt Foundation, and the Japan Society for the Promotion of Science, and expresses gratitude towards these institutions.

PY - 2017

Y1 - 2017

N2 - This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis.We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.

AB - This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis.We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.

KW - Axiomatisations

KW - Ideal elements

KW - Infinitesimal

KW - Intuitionism

KW - Multiverse

KW - Naive integers

KW - Nonstandard analysis

KW - Protozoa

KW - Set-theoretic foundations

KW - Soritical properties

KW - Superstructure

KW - Ultraproducts

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DO - 10.14321/realanalexch.41.1.0193

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SN - 0147-1937

VL - 42

SP - 193

EP - 252

JO - Real Analysis Exchange

JF - Real Analysis Exchange

IS - 2

ER -

Approaches to analysis with infinitesimals following Robinson, Nelson, and others

Abstract

Keywords

ASJC Scopus subject areas

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