Math504: Set Theory

University of Illinois at Chicago

Instructor: Tom Benhamou

My Office: SEO 616

Office hours: Mon 1:00 pm to 2:00 pm

E-Mail tomb at uic .edu

MWF 12:00pm to 1:00pm Addams Hall 310

Textbook: "Set Theory: An Introduction to Independence Proofs" by K.Kunen

Course Website: homepages.math.uic.edu/~tomb/Math504.html

Description

Set theory plays several important roles in the mathematical landscape. First, it lays the formal foundations for most mathematical theories. Secondly, set theory is an attempt to quantify the infinite and to rigorously treat infinite objects. During the first part of this course, we will develop the formal axiomatic set theory ZFC and present the mathematical universe from the point of view of modern Set theory. The investigation of the infinite and the attempt to formalize it, led mathematicians such as Cantor, Suslin, Lebasgue and others to some intriguing and fascinating problems. Some of these problems were traditionally resolved, while other were settled in a surprising way - they were provably unsolvable, or more precisely, they were proven to be independent of the axioms of set theory ZFC. Perhaps the most famous one, is the Continuum Hypothesis, but other problems such as the Suslin hypothesis are less known and reveal the nature of some of the most basic objects in mathematics. To establish independence results, we will develop several models of set theory such as the constructible universe of Godel and the forcing method of Cohen and eventually we will present the proof that CH is independent of ZFC. Also, we will discuss cardinal arithmetics in general, and in the context of theories which extends ZFC known as Large cardinals.
The students are assumed to be familiar with naive set theoretic concept such as: basic sets definition, sets operations, relations, functions, orders, equivalence relations and the basic results about cardinalities such as the Cantor-Berstein theorem and the Diagonalization theorem of Cantor. Also familiarity with basic model theory is assumed, the completeness and two incompleteness theorems are assumed to be known by the audience.