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

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

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, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa.

Citation

Fletcher, P., Hrbacek, K., Kanovei, V., Katz, M. G., Lobry, C., & Sanders, S. (2017). Approaches to analysis with infinitesimals following Robinson, Nelson, and others. Real Analysis Exchange, 1 -59. https://doi.org/10.14321/realanalexch.42.2.0193

• Journal Article Type: Article
• Acceptance Date: Oct 28, 2016
• Publication Date: Nov 3, 2017
• Journal: Real Analysis Exchange
• Print ISSN: 0147-1937
• Peer Reviewed: Peer Reviewed
• Pages: 1 -59
• DOI: https://doi.org/10.14321/realanalexch.42.2.0193
• Publisher URL: https://www.jstor.org/stable/10.14321/realanalexch.42.2.0193
• Related Public URLs: https://arxiv.org/abs/1703.00425