# Math 504 Set Theory I

# Spring 2008

Instructor: David Marker

Class: MWF 9-9:50 302 AH

Office: 312 SEO

Office Hours: (in 411 SEO) M 10-11, Th 2-3 and by appointment

phone: (312) 996-3044

e-mail: marker@math.uic.edu

course webpage: http://www.math.uic.edu/~marker/math504

### Description

In the first half of the century it was shown that most of mathematics
can be formalized inside of set theory and that a simple set of axioms
could be given so that every acceptable proof followed formally from these
axioms. Godel's Incompleteness Theorem implies that there are mathematical
truths not settled by these axioms. The most famous is the Continuum
Hypothesis that asserts that there are no infinite sets of cardinality
greater than the natural numbers but less than the real numbers.

This course will start by introducting the axioms for set theory and
developing the basic theory of cardinals and ordinals. We will then
begin looking at models of set theory and prove that the Continuum Hypothesis
is neither provable nor refutable.
The topics covered will include:
- the Zermelo-Frankel axioms for set theory
- the axiom of choice
- ordinals and cardinals
- models of set theory
- Godel's constructible universe and the consistency of CH
- Cohen's method of forcing and the independence of CH

### Texts

K. Kunen, * Set Theory: An Introduction to Independence Proofs*,
North-Holland, 1979.

There are more elementary books that might be useful for some
of the begining material in the course.
- K. Hrbacek and T. Jech,
*Introduction to Set Theory *
- K. Devlin,
* The Joy of Sets*
- A. Levy,
* Basic Set Theory*

### Prerequisites

Graduate standing. Familiarity with basic concepts from logic: languages, models,
Godel Incompleteness and the Lowenheim-Skolem Theorem.
### Grading

I will give out about 8 problem sets. You may work together on homework problems
(and I encourage you to do so), but when you turn in the problem you should acknowledge that
you have worked together.
I reserve the right to give an in-class final exam.
### Assignments

Dave Marker's Home Page

### Other links

Week-by-Week Syllabus
Construction of many almost free abelian groups

Last updated 4/16/08