SET THEORY

 

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.

 

Course Text:

 

Intermediate Set Theory, by Drake and Singh 

 

Assessment:

 

Exercise sets, one mid term, and one final examination.

 

Course Schedule:

 

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

 

 

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