Topic: set theory versus category theory as the foundation of mathematics
Seminar leaders: John Burgess and Hans Halvorson
NEWS: Here is a link to the paper that Ieke Moerdijk will present. www.jstor.org/stable/2586617. If you want to look under the hood, to see the topos theoretic tools that led to the proof, look here: doi:10.1016/S0022-4049(97)00107-2.
Meeting time: Tuesdays 10:00am to 12:50pm.
Seminar description
The six weeks devoted to set theory will be concerned with the question, "In what sense (if any) does set theory serve as a 'foundation' for contemporary mathematics?" The question will be approached through an historical examination of how set theory came to occupy its present position (however one wants to characterize that position). No substantial previous knowledge of axiomatic set theory will be presupposed. (A substantial part of these six weeks will be devoted to lectures introducing the relevant material as needed.) Special attention will be given to the issue of "structuralism" in mathematics. A syllabus for this part of the seminar will appear shortly at www.princeton.edu/~jburgess.
The second six weeks of the course will discuss the role of category theory (and more specifically of topos theory) in the foundations of mathematics. Does category theory provide an alternative to set theory as a foundation of mathematics? If so, what are the advantages and disadvantages of a category-theoretic approach as opposed to the traditional set-theoretic approach? We will focus on recent arguments to the effect that category theory supplies the tools for a structuralist interpretation of mathematics. A syllabus for this part of the seminar will appear shortly at www.princeton.edu/~hhalvors/teaching.
Reading list for category theory part of the seminar [pdf]
Guest speakers: Steve Awodey (Mar 29), Ieke Moerdijk (Apr 26), Richard Pettigrew (TBA)