Home Page for Math 4LT3/6LT3

Topics in Logic: Axiomatic Set Theory

2012-13

Course content

We will start with constructing the reals and the natural numbers, to see how these fundamental mathematical concepts can be understood using sets. Then we'll turn to the axiomatic set-up for set theory, and see how very few axioms are needed to build the complicated set-theoretic objects we are used to dealing with. Returning to the natural numbers, we'll prove that definition by recursion gives a set which is a function, and hence show how to define arithmetic on the natural numbers. The notion of equinumerous gives us a sense of sizes of infinity, with and without an ordering (ordinals and cardinals). We see a need for the Axiom of Choice, and we'll discuss some it's surprising consequences in other parts of mathematics. Depending on time available, and the interest of the class, we can look at Ramsey's theorems, non-well-founded set theory or large cardinals. We will finish with a discussion of consistency and the continuum hypothesis.

Textbook

"Introduction to Set Theory" Karel Hrbacek and Thomas Jech, 3rd edition, CRC Press

Another useful reference: "Classic set theory for guided independent study" Derek Goldrei, Chapman and Hall

Course structure

The course meets three times per week: M 9:3--10:20, T 10:30-11:20 and Th 9:30-10:20. I plan to lecture about once per week; the other two hours will have students presenting lemmas and exercises that move the theory forward. In general, I expect to introduce a topic on a Thursday, giving you four days to prepare presentations for Monday and Tuesday. Topics will be assigned (and posted on the calendar below) each Thursday. There will also be further homework problems posted, and anyone might be asked to present one of these. For each subsection of the text that we cover, two students will be assigned to write up the class solutions to the exercises. In the last week of class, there will be a take-home final exam. Graduate students registered for 6LT3 will also be required to write a short discussion on an advanced topic in set theory.

Assessment

Class participation and presentations: 50%
Written exercises: 25%
Final exam: 25%

Announcements  

28 March 2013 Tour of the Russell archive is on for Thursday April 4 at 9:30. Let's meet just inside Mills Library.

23 March 2013 There is no final exam for this course, but there will be a final homework set. So far, we have seen the fundamentals of axiomatic set theory. For the last homework assignment, I would like you to read one of the later chapters of the textbook, which give an idea of what advanced set theory and contemporary research in set theory is all about. So choose one of the following reading and problem sets. Your final submission is due by noon on  Wednesday, April 10 (last day of classes).
Chapter 11: read 11.1, problems 1.3, 1.7
                    read 11.2 problems 2.2
                    read 11.3 problems 3.2, 3.6, 3.7
                    read 11.4

Chapter 12: read 12.1 problems 1.1, 1.4
                    read 12.2 problems 2.1, 2.2
                    read 12.3 problem 3.3
                    read 12.4 problem 4.4

Chapter 14: read 14.1 problems 1.3, 1.4
                    read 14.2 problems 2.2, 2.5
                    read 14.3 problems 3.4, 3.5

16 January 2013 My office hours this semester are Monday 10:30-11:30, Tuesday 9:30-10:30 and Thursday 13:30-14:30.

1 January 2013 Welcome to Math 4LT3/6LT3! The first week of class will be taught by Dr Speissegger. I will return (from a research conference in Germany) in time for the second week.

Course Calendar 

Subject to change throughout the semester.

Week
 Topic
 Presentations
 Remarks
Week 1 Jan 7 - 11
 Definition of the real numbers as Dedekind cuts
read about the reals as Cauchy sequences
Goldrei: 2.13, 2.19
Week 2 Jan 14 - 18
 Definition of the natural numbers
Induction and the ordering on N
First axioms of set theory

Week 3 Jan 21 - 25
 Presentation: Theorem 3.2.2 N is linearly ordered
Presentation: Theorem 3.2.4 N is well-ordered
Axioms for set theory: relations, functions, orderings
 Kris, Emily, Chelsea;
Mohdeep
 Exercises: 3.1.1, 3.2.2, 3.2.3, 3.2.7, 3.2.8, 3.2.11, 3.2.13

Scribes: Adam, Josh L.
Week 4 Jan 28 - Feb 1 Presentation: Lemma 2.3.11
Presentation: Theorem 2.3.12
Presentation: Theorem Exercise 2.3.9
The recursion theorem: arithmetic  		Se-Jin
Josh L.
Adam
Exercises: Chapter 2: 1.6, 2.3, 3.6, 3.7,

Scribes: Nigel P., Sean
Week 5 Feb 4 - 8
 Presentation: addition is associative
Presentation: multiplication is commutative
Presentation: mult distributes over addn
Cardinality, Schroder-Bernstein, pigeonhole principle 		Nigel S.
Sean
Nigel P.
 Exercises: think about how to formulate and prove the statements for presentation, and also Ex 4.7 from chapter 3.

Scribes: Chelsea and Mohdeep
Week 6 Feb 11 - 15
 |AxB|=|BxA|
|A^2|=|AxA|
|Ax(B\cupC)|=|AxB\cupAxC|
|A^(B\cupC)|=|A^BxA^C|
|C^AxB|=|(C^B)^A|
comparing cardinalities of infinite sets 		Mohdeep

Emily

Kris
 Exercises: read chs 4.1 and 4.2
From Chapter 4: 2.1, 2.2, 2.3, 2.4, 2.5

Scribes: Se-Jin and Nigel S.
Reading Week Feb 18 - 22


Week 7 Feb 25 - Mar 1
 ch 4 Lemma 2.2
ch 4 Corollary 2.2
Cardinals, the continuum hypothesis		 Nigel P.
Josh
 Homework problems given in class

Scribes: Kris, Emily
Week 8 Mar 4 - 8
 Ch 4 Ex 4.4
Ch 4 Ex 4.6
Well-ordered sets and ordinals 		Sean
Mohdeep and Nigel S.
 Read Chapter 4, section 4 on Linear Orderings
Problems from Ch 4: 4.2, (4.4, 4.6), 4.10, 4.11, 4.12

scribes: Adam, Josh
Week 9 Mar 11 - 15
 Thm 6.2.6 a, b, c
Thm 6.2.6 d
Lem 6.1.4
Cor 1.5
Axiom of replacement; transfinite induction
 Chelsea
Se-Jin
Kris
Nigel S.
 Read chapter 6, sections 1 and 2
1.1, 1.2, 1.5
2.4, 2.7, 2.8

scribes: Nigel P. Sean, Mohdeep
Week 10 Mar 18 - 22
 Thm 6.5.3
Thm 6.5.8
Initial ordinals and alephs
 Josh
Nigel P.
 Read Theorem 6.4.5, and sections 6.4, 6.5
Problems: 6.3.3, 6.3,4, 6.5.4, 6.5.5, 6.5.10
scribes: Chelsea, Emily, Se-Jin
Week 11 Mar 25 - 29
 Thm 7.1.9 (b)
Axiom of choice 		Adam
 Read chapter 7, section 1 and 2
Problems: 7.1.6, 7.1.5, 7.2.5, 7.2.6
scribes: Nigel S, Kris
Week 12 Apr 1 - 5
Axiom of choice
Final homework set
Week 13 Apr 8 - 10
 Discussion of consistency