Course information
Instructor: Prof. John Hunter
Lectures: MWF 4:10–5:00 a.m., Bainer 1130
Office hours: F 12:30–1:30 p.m., R 1:10–2:15 p.m.
CRN: 53308
University of California
Davis, CA 95616, USA
e-mail: jkhunter@ucdavis.edu
Office Phone: (530) 554-1397
Office: Mathematical Sciences Building 3230
Course Information
Georg Cantor originated modern set theory in 1874 with a proof that the real numbers are uncountably infinite and have different cardinality from the countably infinite integers. One of the first questions Cantor asked, in 1878, was if there any any cardinal numbers strictly between those of the integers and the real numbers. The conjecture that there are no such cardinal numbers is called the continuum hypothesis.
Cantor attempted throughout the rest of his life to resolve this question without success, and it was the first problem in the famous list of 23 unsolved problems in mathematics proposed by Hilbert in 1900. The problem was answered in an unexpected way. In 1940, Kurt Godel showed that the continuum hypothesis cannot be disproved from the standard, and apparently complete, ZFC axioms of set theory; and in 1963, Paul Cohen showed that the continuum hypothesis cannot be proved from those axioms either. This independence result raises deep questions about our conception of sets and the infinite and the possibility of fully capturing mathematical concepts by finite systems of axioms.
In this class, we will develop axiomatic ZFC set theory, explain some of the ideas behind the independence results of Godel and Cohen, and discuss their implications for the foundations of mathematics. Set theory is a fairly self-contained subject. The main prerequisite for the class is an appreciation of axiomatic systems and the ability to carry out rigorous mathematical proofs. For example, MAT 25 (Advanced Calculus) and/or MAT 108 (Introduction to Abstract Mathematics) should provide a sufficient background.
Announcement
Problem set 4 was the last assignment for the class, so have a good summer. (There will be no in class final.)
The last day of class will be Mon, Jun 1. There will be no class on Wed, Jun 3.
Some more about forcing is in this paper by Timothy Chow: A beginners guide to forcing.
Text
Classic Set Theory, Derek Goldrei, Chapman & Hall/CRC, 1998.
Homework
Problem numbers refer to the exercises in the text.
Set 1 (Mon, Apr 6)
Read Chapter 1 (Outline and assumed knowledge) of the text.
-
Set 2 (Mon, Apr 13)
Exercise 3.2, p. 34
Exercise 3.3, p. 38
Exercise 3.5, p. 42
Exercise 3.8, p. 45
Exercise 3.10, p. 46
Exercise 3.11, p. 47
Exercise 3.13, p. 47-
Set 3 (Mon, May 4)
Exercise 4.5, p. 74
Exercise 4.12, 4.13 p. 81
Exercise 4.19, 4.21, p. 84
Exercise 4.31, p. 89
Exercise 4.32, p. 90
Exercise 4.41, p. 95
Exercise 4.45, p. 100-
Set 4 (Mon, May 25)
Exercise 7.11, p. 170
Exercise 7.34, p. 183
Exercise 7.44, p. 189
Exercise 7.47, p. 190
Exercise 7.59, 7.60, p. 197
Exercise 8.11(b), p. 210
Exercise 8.16, p. 216
Exercise 8.22, p. 216
-
Set 4 (Mon, May 25)
-
Set 3 (Mon, May 4)