MATH 6300 CARDINAL AND ORDINAL NUMBERS

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

Syllabus:

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.