## 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 Zorn’s 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 Cohen’s Independence results)

**Prerequisites:
**
Graduate standing.