Math 582 - Introduction to Set Theory (Winter 2021)

Instructor / Office hours

Dr. Iian Smythe (smythe@umich.edu) / Tuesday 3:00pm-4:30pm on Zoom.

Lecture (remote, synchronous)

TTh 1:00pm-2:30pm on Zoom.

Textbooks & Resources

All material (including Zoom links, homework, recorded lectures, and lecture notes) will be posted on the course Canvas site.

Required textbook: Herbert B. Enderton, Elements of Set Theory, Academic Press, 1977. Freely available through University of Michigan Libraries: bit.ly/2LEuGbH.

Course outline

An introduction to axiomatic set theory, the foundations of mathematics, and the study of the infinite. Our course is divided into two parts:

• Set theory as foundations (Ch. 1-5 in Enderton): The Zermelo-Fraenkel axioms of set theory, algebra of sets, functions, relations, and the construction of number systems. We will see how set theory can be used as a common axiomatic foundation for all of modern mathematics.

• Set theory proper (Ch. 6-9 in Enderton): Cardinals (“sizes” of infinity), ordinals (“lengths” of infinity), transfinite methods, the Axiom of Choice, and the Continuum Problem. We will learn how to compute and compare sizes of sets across mathematics, develop tools for generalized (potentially uncountable) inductive constructions, and obtain a global picture of the set-theoretic universe.

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