2005/06 Undergraduate Module Catalogue

MATH3123 Set Theory

15 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

Pre-requisites

MATH2040 or equivalent.

This module is approved as an Elective

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

Lectures: 26 hours. 7 examples classes.

Methods of assessment

One 3 hour examination at end of semester (100%).

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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