# Elements of set theory, fall 2009

### Lecturer

Lauri Tuomi

### Scope

10 cu.

### Type

Intermediate studies.

### Description

Set theory is widely accepted as the foundation for mathematics, and serves as the tool for the rigorous study of infinite collections. We will go through the set theoretical construction of real numbers and the basic theory of transfinite ordinal and cardinal numbers. Though we will mainly focus on the so-called naive approach to set theory, the axiomatic approach will not be wholly omitted. In particular, we will take a look at various forms of the somewhat controversial Axiom of Choice.

### Prerequisites

No special prerequisites.

### Lectures

Weeks 37-43 and 45-51, Monday 12-14 and Tuesday 12-14 in room B322. Two hours of exercise classes per week. First lecture on Tuesday 8.9.

### Exams

The exam takes place **Tuesday 15.12.** at **12-16** in **Room BK107.**

An alternative is to take the exam in the department examination 22.12. Please inform the lecturer if you need to take this latter exam (or if you cannot take neither.)

### Bibliography

Lectures will mostly follow H. Enderton's Elements of set theory.

### Registration

Did you forget to register? What to do.

### Exercise groups

Group |
Day |
Time |
Place |
Instructor |
---|---|---|---|---|

1. |
Friday |
14-16 |
D123/B322 |
Kaisa Kangas |

The exercise class is in room D123 except that 27.11. it is in B322. There are model solutions by K.K. in the course file.

##### List of exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4 Note: Corrected version, an extra assumption was added to Problem 3.

Exercise 5

Exercise 6

Exercise 7

Exercise 8 Corrected 16/11: an assumption was added to 1(d) and a restriction was added to 2.

Exercise 9

Exercise 10

Exercise 11

Extra: exercise 12

### Schedule of lectures

**WEEK 37:** We considered motivational aspects to set theory. Chapter 1 of Enderton's.

**WEEK 38:** Basic axioms and operations, relations. Chapter 2.

**WEEK 39:** Relations: functions, equivalences, orderings. Chapter 3.

**WEEK 40:** Construction of natural numbers; recursion, arithmetic. Chapter 4.

**WEEK 41:** More on natural numbers: arithmetic, order. Construction of integers. Chapters 4-5.

**WEEK 42:** Structure of integers: arithmetic, order. Construction of rational numbers. Chapter 5.

**WEEK 43:** More on rational numbers. Construction of real numbers. Chapter 5.

**WEEK 44:** Fall semester break. No lectures nor exercises.

**WEEK 45:** We will complete the study of real numbers and begin studying the sizes of sets. Chapters 5-6.

**WEEK 46:** Cardinal arithmetic. Ordering the cardinalities. Chapter 6.

**WEEK 47:** Cardinal arithmetic and the Axiom of Choice. Chapter 6.

**WEEK 48:** Absorption Law of Cardinal Arithmetic. Transfinite recursion. Chapters 6-7.

**WEEK 49:** Ordinal numbers. Chapter 7.

**WEEK 50:** Depts paid: definition of cardinal numbers and the proof of Zorn's Lemma and Well-Ordering Thm. Rank and the cumulative hierarchy. Chapter 7.

**WEEK 51:** Monday: recapitulation, miscellaneous results. Tuesday: Exam. Friday: extra exercise session.