Hyperreals and Their Applications (Part 2)

Sylvia Wenmackers (Groningen) gives the second 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). In section 4, we give an overview of the areas of applications of NSA. We return to a selection of them in the subsequent sections. Section 5: history of the calculus; section 6: intuitions about infinitesimals; section 7: paradoxes of infinity; section 8: probability and formal epistemology; and section 9: physics and philosophy of science.