Updated: April 9, 2007
The theory of Ordinals and Cardinals will be developed
within Zermelo Fraenkel Set Theory. The course is meant for students who wish to
gain a firm understanding of the set theoretic foundations of mathematics. Thus,
our approach will be strictly axiomatic. A few tools from mathematical logic
will be developed within the course. The course is based on my
Notes on Set Theory
But I also recommend the book:
Karel Hrbacek, Thomas Jech, Introduction to Set Theory, Third Edition, Revised and Expanded, Taylor & Francis, ISBN-10:0-8247-7915-0
1. The Zermelo Fraenkel Axioms of Set Theory (and a mini course in logic).
2. Ordinals (contains a proof of the general recursion theorem).
3. The Axiom of Choice (with Zornís Lemma and ordinal arithmetic).
4. The Axiom of Foundation (important for understanding the ZF-Hierarchy of sets and the concept of rank)
5. Cardinals (Cantor-Bernstein, cardinal arithmetic; discussion of the GHC and of Cohenís Independence results)Prerequisites: Graduate standing.