The main objectives of the course are to give you a preliminary sense of what academic mathematics is, to develop fluency with mathematical language and grammar, and to build confidence with mathematical reasoning. As a foundational field of pure math, set theory provides a suitable avenue for your first foray into rigorous mathematics, and holds numerous abstractly beautiful and unexpected results as a delightful bonus.
The course will begin with an introduction to propositional and predicate logic and to formal languages. We will then introduce the axioms and basic definitions of Zermelo-Fraenkel-Choice Set Theory one-by-one and examine their corollaries, including Cantor’s theorem and the Schröder-Bernstein theorem. Next, we will focus on ordinal numbers and their properties, including ordinal arithmetic. The final portion of the course will be driven by student interests: possible topics include a discussion of proof theory and model theory or an examination of large cardinal axioms.
Class time will be divided between lectures, during which new concepts and definitions will be explained and examples are given, and group proof-writing exercises, in which you will help each other think through the proofs of various propositions. You will be able to consult the instructor for clarification and hints, and will eventually share your discoveries with the rest of the class. Homework will not be collected and graded; instead, you will be given exercises of varying degrees of difficulty to think about in order to build greater familiarity with the material. You will not be required to purchase any textbooks. We will sometimes refer to Willaim A. R. Weiss’ An Introduction to Set Theory, which is freely available on the author’s webpage; this book will be supplemented by lecture notes written by the instructor.
At the conclusion of the course, you will:
• have the necessary skills to independently read and understand introductory texts on a range of topics in abstract mathematics, and;
• develop confidence in your ability to formulate and carefully write rigorous mathematical proofs.
Because set theory is foundational, no background 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 register. The work of the course will involve manipulating sequences of symbols that represent abstract, imaginary logical structures. All the rules for these manipulations will be taught in the course, but it will be best if students do not have an aversion to this kind of work.
Online sections of Pre-College courses are offered in one of the following modalities: Asynchronous, Mostly asynchronous, or Blended. Please review full information regarding the experience here.