Time & Location

Professor

Office hours

Mon, Wed, Fri: 10-11, 12-1
1096 East Hall
Kenneth Harris
homepage: kaharris.org
email:
office: 1842 East Hall
tel: 734.763.4703
Mon, Wed, Fri: 1-2
(and by appointment)

 

[ Syllabus ] [ Schedule ]


 

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 1.1-2,1.4
Notes on Naive Set Theory
  1. History of Set Theory
  2. Wed Notes
  3. Fri Notes
HW1 (part 1)
solution
Week 2 (Jan 12 - 16)
Naive Set Theory: Functions
Finite and Countable sets
Notes on Naive Set Theory
  1. Mon Notes
  2. Wed Notes
  3. Fri Notes
HW1 (part 2)
solution
Week 3 (Jan 19 - 23)
Introduction to Cardinals: Counting Infinities
Uncountable sets
Trouble in paradise
Chapter 4.1-3
  1. Wed Notes
  2. Fri Notes
HW2 (part 1)
solution
HW1 due (Jan 23)
Week 4 (Jan 26 - 30)
Introduction to the Axioms Chapter 1.3-4, 2.1
  1. Axioms of Set Theory
  2. Mon Notes
  3. Wed Notes
  4. Fri Notes
HW2 (part 2)
solution
Week 5 (Feb 2 - 6)
Discrete Mathematics -- Relations, functions, order Chapter 2.2-2.5
  1. Mon Notes
  2. Wed, Fri Notes
HW3 (part 1)
solution
HW2 due (Feb 6)
Weeks 6-7 (Feb 9 - 20)
The natural numbers Chapter 3.1-4
  1. Mon (1) Notes
  2. Wed (1), Fri (1) Notes
  3. Mon (2) Notes
  4. Wed (2) Notes
HW3 (part 2)
solution
HW3 due (Feb 20)
 
Spring Break (Feb 21 - March 1)
 
Week 8 (March 2 - 6)
Ordinal numbers Chapter 6.1-3
  1. Mon, Wed Notes
  2. Fri Notes
HW4 (part 1)
solution
HW4 (part 2)
solution
Week 9 (March 9 - 13)
Transfinite recursion and ordinal arithmetic Chapter 6.4-5
  1. Basic properties of ordinal arithmetic
  2. Mon Notes
  3. Wed Notes
  4. Fri Notes
HW4 due (March 13)
Week 10 (March 16 - 20)
Well-ordered sets and Ordinals Chapters 6.1
  1. Mon Notes
  2. Wed Notes
  3. Fri Notes
HW5 (complete -- due March 27)
solution
Week 11 (March 23 - 27)
Power Set and Cardinal number Chapters 4.1-4.3, 5.1, Chapter 7
  1. Mon Notes
  2. Wed Notes
  3. Fri Notes
HW6 (complete -- due April 10)
HW5 due (March 27)
Week 12 (March 30 - April 3)
Axiom of Choice and equivalents Chapter 8
  1. Mon Notes
  2. Wed Notes
  3. Fri Notes (Cardinal Exponentiation)
HW7 (Part I, due April 20)
Week 13 (April 6 - 10)
Cardinal exponentiation Chapter 9
  1. Mon Notes
  2. Wed Notes
  3. Fri Notes
HW6 due (April 10)
HW7 (Part II, due April 20)
Week 14 (April 13 - 17)
Axiom of Foundation
Silver's Theorem
Chapter 14.1-14.2
Chapter 11
  1. Mon Notes
  2. Wed Notes
Week 15 (April 20)
HW7 due (April 20)