Math 682 - Set Theory (Fall 2021)

Instructor / Office hours

Dr. Iian Smythe (smythe@umich.edu) / By appointment.

Lecture

TTh 1:00pm-2:30pm in 242 West Hall.

Textbooks & Resources

All material (including problems, lecture notes, and other resources) will be posted on the course Canvas site.

Textbook: Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, Elsevier, 1980. This text is freely available from the University of Michigan Library.

Course outline

In this course, we will study methods for showing that certain mathematical statements are independent (i.e., neither provable nor disprovable) from the usual Zermelo–Fraenkel axioms of set theory, giving concrete examples of the incompleteness phenomenon originally discovered by Gödel. Along the way, we will cover some of the background necessary for doing research in set theory today.

We will begin with a review of axiomatic set theory at the level of Math 582: the ZFC axioms, ordinals, cardinals, the universe of sets, and well-founded relations. We will then discuss models of set theory and Gödel’s constructible universe (known as “L”), where both the Continuum Hypothesis and the Axiom of Choice are true.

The second half of the course will be devoted to Cohen’s method of forcing, which builds extensions of models of set theory, and will be used to show that both the Continuum Hypothesis and the Axiom of Choice may fail, establishing their independence. If time allows, we will discuss further applications of these methods, such as the consistency of Martin’s Axiom (an alternative to the Continuum Hypothesis), and the construction of inner models where all sets of reals have strong regularity properties (e.g., measurability).

See the course syllabus (PDF, Google Docs) for more detailed information on course format, evaluation, and expectations.