Time & Location 
Professor 
Office hours 




Syllabus
Prerequisites
Math 412 or 451 or equivalent experience with abstract mathematics. No course in mathematical logic is presupposed, however you are expected to be familiar with methods of proof that are developed in abstract mathematics classes such as analysis and algebra.
Text
Karel Hrbacek and Thomas Jech, Introduction to Set Theory (3rd ed.),
CRC, 1999.
Synopsis
All mathematical concepts (such as function, relation, group, topological space) and mathematical objects (the real number π, the collection of all real numbers, the Euclidean plane) can be defined using the primitive notions of set and set membership. One of the goals of this course is to develop some understanding of how Set Theory plays this role. At the same time, many new concepts will be introduced (e.g. transfinite ordinal and cardinal numbers, the Axiom of Choice, Zorn's Lemma) which play a major role in many branches of mathematics.
We will cover the first nine chapters of the text. Time permitting, we will continue with more advanced material from Chapters 11 to 14.
Grading
There will be no exams in this course. Grading will be based upon regular homework (I expect about seven assignments).
This is a proof intensive course. Scoring homework problems will be based
on the following five point evaluation scheme (scaled in multiples of five
depending on the difficulty of the proof):
Score  Evaluation 
5  Proof is correct as written. 
4  Missing minor point or an incorrect reason given for a justification. 
3  Missing a major point that leaves the proof well short of its goal. 
2  Proof goes in a totally wrong direction. 
1  Almost no discernible content. 
Schedule
Topic  Readings  Notes  Homework 

Week 1 (Jan 5  9)  
Naive Set Theory: Basic assumptions and set operations.  Chapters 1.12,1.4 Notes on Naive Set Theory 
HW1 (part 1) solution 

Week 2 (Jan 12  16)  
Naive Set Theory: Functions Finite and Countable sets 
Notes on Naive Set Theory 
HW1 (part 2) solution 

Week 3 (Jan 19  23)  
Introduction to Cardinals: Counting Infinities Uncountable sets Trouble in paradise 
Chapter 4.13 
HW2 (part 1) solution HW1 due (Jan 23) 

Week 4 (Jan 26  30)  
Introduction to the Axioms  Chapter 1.34, 2.1  HW2 (part 2) solution 

Week 5 (Feb 2  6)  
Discrete Mathematics  Relations, functions, order  Chapter 2.22.5 
HW3 (part 1) solution HW2 due (Feb 6) 

Weeks 67 (Feb 9  20)  
The natural numbers  Chapter 3.14 
HW3 (part 2) solution HW3 due (Feb 20) 

Spring Break (Feb 21  March 1)  
Week 8 (March 2  6)  
Ordinal numbers  Chapter 6.13 
HW4 (part 1) solution HW4 (part 2) solution 

Week 9 (March 9  13)  
Transfinite recursion and ordinal arithmetic  Chapter 6.45  HW4 due (March 13)  
Week 10 (March 16  20)  
Wellordered sets and Ordinals  Chapters 6.1 
HW5 (complete  due March 27) solution 

Week 11 (March 23  27)  
Power Set and Cardinal number  Chapters 4.14.3, 5.1, Chapter 7 
HW6 (complete  due April 10) HW5 due (March 27) 

Week 12 (March 30  April 3)  
Axiom of Choice and equivalents  Chapter 8  HW7 (Part I, due April 20)  
Week 13 (April 6  10)  
Cardinal exponentiation  Chapter 9  HW6 due (April 10) HW7 (Part II, due April 20) 

Week 14 (April 13  17)  
Axiom of Foundation Silver's Theorem 
Chapter 14.114.2 Chapter 11 

Week 15 (April 20)  
HW7 due (April 20) 