Set Theory

Department: 
MATH
Course Number: 
8803-BER
Hours - Lecture: 
3
Hours - Total Credit: 
3
Typical Scheduling: 
Not Regularly Scheduled

This course will provide a rigorous introduction to the modern area of (axiomatic) set theory.

Prerequisites: 

Beyond knowing what words such as “bijection” and “countable” mean, no prior familiarity with set theory or logic will be assumed.

Course Text: 

There will be no required textbook. Lecture notes and other reading material will be provided.

Topic Outline: 

Here are some topics and questions we will touch on.

  • The generally accepted set of axioms for mathematics are the so-called Zermelo–Fraenkel axioms with the Axiom of Choice, or ZFC for short. What are they? Why were these particular axioms chosen? Why do we think they suffice for mathematical practice?
  • What makes the Axiom of Choice so notorious? What are some of its counter-intuitive consequences? What would math be like if the Axiom of Choice didn’t hold?
  • What are classes? Many mathematicians have heard that a class is a collection that is “too large” to be a set, but what does that mean exactly?
  • We will learn about ordinals and transfinite induction, which is a powerful proof technique for dealing with infinite sets that works with just one element at a time.
  • The “sizes” of infinite sets are measured by objects called cardinals. What are they? How can we compute the cardinality of a given set? Is it always possible?
  • The Continuum Hypothesis, or CH for short, is a famous problem posed by Georg Cantor in 1878. Remarkably, CH is independent of ZFC, i.e., it can neither be proved nor disproved within the generally accepted mathematical framework. Why is CH so important? What are some of its consequences?
  • How can one prove that a statement—such as CH—is independent of ZFC? (We will definitely show that CH can’t be disproved in ZFC. If we have time, then we’ll discuss why it can’t be proved either.)
  • If all of mathematics can be formulated inside set theory, then this includes set theory itself, right? What would that mean? How can we avoid self-referential paradoxes when talking about set theory inside set theory?
  • Are there special cases of statements such as CH that we can prove?