# 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.

• 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.