Relationships between Constructive, Predicative and Classical Systems of Analysis

Abstract

Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular “impredicative” definitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to classical, set-theoretically based mathematics until the 1960s. Now we have a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis.

This is the last of my three lectures for the conference, Proof Theory: History and Philosophical Significance, held at the University of Roskilde, Denmark, Oct. 31-Nov. 1, 1997. See the first footnote to the first lecture, “Highlights in Proof Theory” for my acknowledgements.