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.
"Approaches to Analysis with Infinitesimals Following Robinson, Nelson, and Others." Real Anal. Exchange 42 (2) 193 - 252, 2017. https://doi.org/10.14321/realanalexch.42.2.0193