Set Theory: The Dazzling Foundation of Abstract Mathematics

Course Description

The premise of set theory is deceptively simple: it is the study of boxes ("sets") and the things they hold inside them ("elements"). Axiomatic set theory is to other areas of mathematics as an operating system is to other computer programs. While it often exists in the background, only obliquely referred to, set theory supplies a solid foundation for our castles in the sky. It gives us a language with which to develop the rest of mathematics.

But set theory is more than foundation for other fields; it opens many dazzling frontiers of thought all on its own. A first example is the discovery that there is a whole menagerie of species of infinity, some larger than others, with all sorts of properties of their own. Just as we can ask which counting numbers are "composite" (i.e. the product of smaller numbers), we can also ask, for instance, which types of infinity can be built from smaller infinities and which cannot.

This course will introduce students to the fundamental ideas and techniques of set theory as well as some of the most accessible examples of surprising and even non-intuitive results in the field.

Although set theory is a field of pure, abstract mathematics, which students typically do not encounter until university, set theory's status as foundational means that no prerequisite knowledge whatsoever is required. Facility for logical reasoning is important, but the primary goal of the course is to develop this skill. This course is a good choice for students who enjoy logic puzzles.

The collaborative discovery of mathematical truth is the most exhilarating experience one can have in mathematics and is also the best way to learn math. For this reason, class time will be split between lecture to convey the key concepts and to illustrate them with examples, Q&A, and closely supervised group work, with an emphasis on the latter.

The text for the course will be course notes written by the instructor which draw from several textbooks in the field. These notes will provide useful review material for students as well as optional problems to help students solidify their understanding of the material outside of class.

The course will begin with an introduction to the academic discipline of mathematics and such notions as axiom, proof, and theorem. The course will then introduce the axioms of Zermelo-Fraenkel set theory. The first week will culminate in a careful proof of Cantor's Theorem. The second week of the course will introduce students to ordinals and the well-ordering theorem as well as such properties of cardinals as regularity and accessibility. The course will conclude with an introduction to large cardinal axioms, time-permitting.

Prerequisites: No knowledge is required for the course. One of the main goals of the course is to convey that abstract mathematics is fun, exciting, and accessible. But the course will be highly abstract from early on, so it will be best for students not to harbor a pre-established fear of or prejudice against abstract mathematics which would have to be overcome. Curious students with no prior exposure to higher math but who are eager to learn are encouraged to apply.