This course is an introduction to set theory. We begin with the naive conception of sets, showing how and why the naive conception leads to several paradoxes, including Russell's paradox. We then develop axiomatic set theory, using the Zermelo-Fraenkel axioms. We will explore how to do mathematics from within set theory and discuss whether set theory serves as a foundation for mathematics. We will show that the Axiom of Choice and the Continuum Hypothesis are independent from the standard Zermelo-Fraenkel axioms, and conclude by looking at the recent “multiverse” interpretation of set theory.
Intermediate Set Theory, by Drake and Singh
Exercise sets, one mid term, and one final examination.
Week 1: Naive set theory and the paradoxes
Week 2: First-order logic
Week 3: The language of set theory
Week 4: Axiomatic set theory
Week 5: Cardinals
Week 6: Order
Week 7: Doing mathematics within set theory
Week 8: The Axiom of Choice
Week 9: The Independence of the Axiom of Choice
Week 10: The Continuum Hypothesis
Week 11: The Independence of the Continuum Hypothesis
Week 12: The set-theoretic multiverse