## MATH3123 Set Theory

### 15 creditsClass Size: 100

Module manager: Professor M. Rathjen
Email: rathjen@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH2040, or equivalent.

### This module is mutually exclusive with

This module is approved as an Elective

### Module summary

Set theory arose out of Cantor's work on Fourier series when he used transfinite ordinals to describe certain sets of real numbers. He then went on to develop a theory of transfinite cardinals. To meet some of the criticisms Cantor's theory met, Zermelo invented a set of formal axioms, later augmented by Fraenkel, which have proved adequate for the development of set theory and most of mathematics. In this module we explain these ideas, develop Cantor's theory and show how it can be incorporated within the framework of Zermelo-Fraenkel set theory.

### Objectives

To present axiomatic set theory as a possible foundation for mathematics, and to illustrate this by constructing standard number systems in a set-theoretical universe. To develop the theory of ordinals and cardinals as far as arithmetical operations on the latter. To introduce simple independence and relative consistency results (independence of axiom of infinity, consistency of axiom of foundation).

On completion of this module, students should be able to:
a) test various abstraction terms for sethood;
b) use axiomatic set theory to set up a foundation for mathematics, including constructions for the natural numbers, and reals;
c) handle elementary arguments involving ordinals and cardinals;
d) recognise and apply some equivalents of the axiom of choice;
e) establish the independence of the axiom of infinity and the relative consistency of the axiom of foundation.

### Syllabus

Two basic motivations for studying set theory are followed up in this module. One is to use it as a foundation for mathematics, and to help us to answers questions such as ' what is a number?' The other is to regard set theory as itself constituting a formal theory (which we hope captures reasonably well the informal notion of 'set'). We seek to understand 'transfinite numbers' originally introduced by Cantor, that is, ordinals and cardinals. The former describe 'how we can count a set' the latter 'how big a set is'. In addition we study the superficially innocuous 'axiom of choice', describe some of its more starting consequences, and discuss Cantor's 'continuum problem' - how many real numbers are there? Some of the questions turn out to be independent of the usual axioms of set theory, and we shall give a flavour of this by describing consistency and independence results in simple cases. The discussion takes place in Zermelo-Fraenkel theory, a system devised early in the twentieth century in an attempt to circumvent the famous 'paradoxes' of Russell, Cantor, and others. The topics covered are: Russell's paradox. The cumulative type structure. The Zermelo-Fraenkel axioms. Von Neumann ordinals. Transfinite induction. Natural numbers as finite ordinals and constructions of the integers, rationals, and reals. Relative consistency of the axiom of foundation. Cardinal numbers and the axiom of choice.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 5 1.00 5.00 Lecture 24 1.00 24.00 Private study hours 121.00 Total Contact hours 29.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated