Michael E. Townsend
Development of axiomatic set theory up to and including the consistency of the Axiom of Choice and Continuum Hypothesis with the Zermelo-Fraenkel Axioms.
Set theory, together with proof theory, model theory, and recursion theory, constitute “mathematical logic.” In this context, “logic” refers to reasoning, while the word “mathematical” has two connotations. On the one hand, the word indicates an interest in the type of reasoning used by mathematicians. To the extent that mathematical logic studies mathematical systems, it is called “meta-mathematics.” The word also indicates that this study employs mathematical techniques, a self-referential feature with many well-known manifestations. The scope of set theory poses a challenge for stand-alone courses. Modern set theory is at least a century and a quarter old, and it contains many sub-branches, each of which is a Ph.D. specialization. Therefore, care must be taken in selecting the material for a single course. As a general matter, this course is organized around four interrelated themes. Many students have seen set theory used as a way to reduce some of the mathematical concepts of traditional mathematics (algebra, analysis, and topology) to even more primitive notions. We illustrate this aspect of set theory by studying the natural number system in set-theoretic terms. Other students have seen set theory used to investigate the transfinite. We illustrate this aspect of set theory by extending the natural number system to include infinite ordinals and cardinals. Fewer students will have studied sets per se. We illustrate this aspect of set theory by considering a few of the issues related to sets containing themselves as members. Finally, we consider the relationship between set theory and mathematics. Roughly speaking, one may take three positions on this relationship: set theory is a part of mathematics; set theory is not part of mathematics, but it is a closely related cognate discipline; set theory is not a part of mathematics, but it is more than a cognate—it is the foundation of mathematics. Given the extensive nature of the field, our treatment of these themes will be somewhat cursory, especially with respect to the last two.
Student learning goals
General method of instruction
Although this course is offered through the philosophy department, we utilize the axiom/definition/theorem/proof method typical of modern treatments of set theory. Therefore, students should have completed a college-level course requiring mathematical proofs, including proofs by induction. At least as important is a certain “mathematical maturity” that is best described as the willingness to study class notes and work through the homework problems. PHIL 470 recommended; graduate students only; undergraduates by permission; no freshmen.
Class assignments and grading
In terms of grading, there will be regular homework assignments (60%) as well as a take home final (40%).