Hyperreals and Their Applications

Sylvia Wenmackers (Groningen) gives the first part of her tutorial "Hyperreals and Their Applications" at the 9th Formal Epistemology Workshop (Munich, May 29–June 2, 2012). Abstract: Hyperreal numbers are an extension of the real numbers, which contain infinitesimals and infinite numbers. The set of hyperreal numbers is denoted by *�R or R*�; in these notes, I opt for the former notation, as it allows us to read the �*-symbol as the prefix 'hyper-'. Just like standard analysis (or calculus) is the theory of the real numbers, non-standard analysis (NSA) is the theory of the hyperreal numbers. NSA was developed by Robinson in the 1960’s and can be regarded as giving rigorous foundations for intuitions about infinitesimals that go back to Leibniz (at least). NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a flavor of model theory. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle. Section 3 explains the ultrapower construction of the hyperreals; it also includes an explanation of the notion of a free ultrafilter.