Spring 2010 Teaching

Math 4730: Set Theory

The first exam has been moved to Friday February 26.

Office hours M 8:00-8:50, W 12:00-12:50, F 10:00-10:50
Class times MWF 9:00-9:50
Class location    ECCR 118
Book Notes handed out in class

First day handout

Lectures

Date
Monday January 11 Introduction.
Basic ideas about logic: truth tables and standard arguments (section introduction)
Wednesday January 13 The first axioms: meaning and some use. (section 1)
the key idea about axioms is that they don't describe what a set is, but what a universe of sets is.
Friday January 15 Axioms, defined properties, and how to write a proof (section 1)
the key idea in writing a proof is often using the standard methods to get down to the hart of the matter, and then reasoning very carefully.
Monday January 18 No class: MLK day.
Wednesday January 20 Ordered pairs and relations (section 2)
The key point was the idea that notions that a priori are not about sets, can be "implemented" in sets. They should then have the main properties of the original notions, but will often also "accidentally" have more properties.
Friday January 22 Functions (section 3, a small start)
We discussed the reason for developing the axioms the way we are, not too strong, not too weak. We also discussed what the axioms are describing.
Monday January 25 Functions (section 3)
Worked some of the theorems in the section emphasizing the role of definitions, that they are not randomly chosen, but need to fit our intuitions. Side effects are to be expected though.
Wednesday January 27 Functions (section 3)
The axiom of choice was introduced and explained.
Friday January 29 replacement procedure (section 3)
Equivalence relations and partitions (section 4)
Monday February 1 Finishing equivalence relations (section 4)
Pictures of relations: easy way of obtaining examples.
Definition and (non-)examples of wellorders (section 5)
Wednesday February 3 Induction: for any well order (section 5) and on N
defining N (section 6)
Friday February 5 Properties of N (section 6):
Induction and some basic properties of our implementation.
Monday February 8 Doing inductive proofs
The Recursion Theorem
Wednesday February 10 Proof of the recursion theorem
Friday February 12 Defining addition.
Properties of addition.
Mon Feb 15-Feb19 Essentially finish Chapter 6
Monday February 22 On the proof of the closure theorem.
Wednesday February 24 On proving existence choice functions from AC.
Friday February 26 First midterm
Monday March 1 The ideas surrounding ordinals and cardinals.
Wednesday March 2 Discussed the exercise on the recursion theorem in great detail.
Friday March 5 Definition cardinals and some properties (Chapter 7)
Monday March 8 Properties of cardinals (Chapter 7)
Wednesday March 10 Union of countably many countable sets.
There are uncountably many reals.
Friday March 12 Characteristic functions (Chapter 7)
Order isomorphisms (Chapter 8)
Monday March 16 Key theorem on ordinals (Chapter 8)
Wednesday March 18 Cardinal arithmetic (Chapter 8)
Friday March 20 No lecture.
Monday March 29 Discussed some homework problems.
Wednesday March 31 Second Midterm Exam
Friday April 2 Basics of Cardinal Arithmetic (Chapter 8):
reason, definitions, robustness.
Monday April 5 Cardinal arithmetic identities: strategies for defining injections or bijections.
Wednesday April 7 More about some injections from Monday.
Working towards K x K = K for inf cardinals:
Decided strategy, and defined and drew the picture of the wellorder.
Friday April 9 Discussed homework problem about eventually constant functions.
Finished K x K = K for inf cardinals.
Started the thinking about computing the cardinalities for closures.
Monday April 12 How to "compute" the cardinality of the closure of a set under partial functions.
Wednesday April 14 Collections that do not form classes.
Induction on well-orders.
Friday April 16 Induction and recursion for ordinals/any well-order.
Monday April 19 Questions
Recursion for ordinals.
Wednesday April 21 Questions
Recursion on the class of ordinals, what is different?
If time: useful picture of the universe.

Homeworks

DueAssignment
Friday January 22
Chapter 1: 1, 2, 5, 6, 7, 11
Friday January 29
Chapter 1: 13
Chapter 2: 1, 2, 6, 7
Chapter 3: 1
Friday February 5
Chapter 1: 12
Chapter 2: 14
Chapter 3: 6, 15, 16, 21
Friday February 12
Chapter 3: 22, 26
Chapter 4: 1, 2
Chapter 5: 1
Friday February 19
Chapter 4: 4, 8
Chapter 5: 2, 6
Chapter 6: 2 (only do (i), (ii), and (v))
Friday March 5
Chapter 5: 5, 8
Chapter 6: 7, 13, 19
Friday March 12
Chapter 3: 24
Chapter 4: 7
Chapter 6: 12, 14, 21, 23
Friday April 2
Chapter 7: 1, 2, 3, 7, 13, 20
Friday April 9
Chapter 7: 16, 21
Chapter 8: 1, 4, 11
Friday April 16
Chapter 7: 6, 15
Chapter 8: 12, 13
Friday April 23:
Chapter 8: 21, 22
Chapter 9: 16, 19

Subjects for Basic Notions Quiz April 19

pdf file is also available.