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

Due

Assignment

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

Truth tables for basic connectives (and/or/if X, then Y/not/iff).
Form of argument by contradiction.
Form of argument by contrapositive.
Form of standard argument to prove there exists x. P(x) .
Form of standard argument to prove for all x. P(x) .
Form of standard argument to prove A => B .
The axioms (up to and including the 8th).
Definition of union of sets (pairs and families).
Definition of intersection of sets (pairs and families).
Definition of {x in A | P(x)} .
Definition of {a,b} .
Definition of {a} .
Definition of the empty set .
Definition of disjoint.
Standard argument for use of disjunction.
Standard argument for equality of sets.
Definition of (a,b) , and its key property.
Definition of Domain.
Definition of Range.
Definition of (binary) relation.
Definition of inverse relation.
Definition of image of a set under a relation.
Definition of function.
Definition of surjection (onto function).
Definition of injection (one-to-one function).
Definition of cartesian product of a system of sets.
Definition of equivalence relation.
Definition of partition.
Definitions of the different properties of orders.
Definition of the principle of complete induction.
Definition of the natural numbers .
Definition of addition, mutiplication, and exponentiation of
Definition of equipotent.
Definition of finite.
Definition of F -closed set.
Definition of {Cl}_F(X) .
Definition of ordinal and cardinal.
The second form of the axiom of choice.
Definition of denumerable and countable.
Definition of order-isomorphism.
Definition of class function.
Definition of cardinal addition, multiplication, and
Definition of the sum of an indexed family of cardinals.

pdf file is also available.