# Set Theory

## Unit aims

To introduce the students to the general theory of sets, as a foundational and as an axiomatic theory.

## Unit description

The aim is to make the course of general interest to students who are not planning to specialize in mathematical logic or Axiomatic Set Theory, but of special interest to those who are.

Set Theory can be regarded as a foundation for all, or most, of mathematics, in that any mathematical concept can be formulated as being about sets. The course shows how we can represent the natural numbers as sets and how principles such as proof by mathematical induction can be seen as being built up from very primitive notions about sets.

We shall see how the pitfalls of the various early "set theoretic paradoxes" such as that of Russell ("the set of all sets that do not contain themselves") were avoided. We develop Cantor's theory of transfinite ordinal numbers and their arithmetic through the introduction of his most substantial contribution to mathematics: the notion of well ordering. We shall see how an "arithmetic of the infinite" can be developed that extends naturally the arithmetic of the finite we all know. We shall introduce the principle of ordinal induction and recursion along the ordinals to extend that of mathematical induction and recursion along the natural numbers. Cantor's famous proof of the uncountability of the real continuum by a diagonal argument, and his revolutionary discovery that there were different "orders of infinity" - indeed infinitely many such - will feature prominently in our basic study of infinite cardinal numbers and their arithmetic.

We shall see how axiom sets can be used to develop this theory, and indeed the whole cumulative hierarchy of sets of mathematical discourse. There will be discussion of the axioms system ZF developed by Zermelo and Fraenkel in the wake of Cantor's work, and about the role the Axiom of Choice plays in set theory.

## Relation to other units

Set Theory may be regarded as the foundation for all mathematics. This course is a prerequisite for Axiomatic Set Theory.

For students interested in the philosophy of mathematics: this course is related to a number of units in the philosophy department in philosophy of mathematics. It should thus be of interest to any joint Maths/Philosophy degree students, and to those on the MA in Logic and Philosophy of Mathematics

## Learning objectives

The student should come away from this course with a basic understanding of such topics as the theory of partial orderings and well orderings, cardinality, ordinal numbers, and the role of the Axiom of Choice. He or she should also have become aware of the role of set theory as a foundation for mathematics, and of the part that axiomatic set theory has to play.

## Transferable skills

The ability to think more deeply about our basic assumptions and concepts.

## Syllabus

- The basic principles and definitions of set theory
- The axiomatic definition of natural number
- Well-ordering and the Axiom of Choice
- Definition by transfinite induction
- Ordinal and cardinal numbers
- The cumulative hierarchy of sets
- Axiomatic Set Theory

## Reading and References

Full Lecture notes will be provided.

- H. Enderton,
*Elements of Set Theory*, Academic Press - D. Goldrei,
*Classic Set Theory*, Chapman & Hall - B. Rotman & G.T. Kneebone,
*The theory of sets and transfinite ordinal numbers*, Oldbourne Mathematical Series. (QA248 ROT.)

The following cover more than we need:

- K. Devlin,
*The Joy of Sets*, Springer - W. Just and M. Weese,
*Discovering Modern Set Theory I*, AMS Graduate Studies in Mathematics, Vol. 8. Is couched in more sophisticated language, Chapters 1-5, 10,11 are sufficient. - E. Schimmerling,
*A course in Set Theory*, CUP, 2011. First 4 chapters only. QA248.SCH

If you are intending to take Axiomatic Set Theory, the following are paperback and are worthwhile purchases:

- F. R. Drake and D. Singh,
*Intermediate Set Theory*, J. Wiley & Co - K. Kunen,
*Set Theory*, College Publications, Studies in Logic, vol 34. Chapter 1 on Background only.

**Unit code:** MATH32000

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Professor Philip Welch

## Pre-requisites

None

## Co-requisites

None

## Methods of teaching

Lectures and Exercise Sheets

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.