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2008/09 Taught Postgraduate Module Catalogue

MATH5123M Advanced Set Theory

15 creditsClass Size: 20

Module manager: Professor M. Rathjen

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

Pre-requisite qualifications

MATH2040 or equivalent.

This module is mutually exclusive with

MATH3123Set Theory

This module is not 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.


On completion of this module, students should be able to:

a) explain the notions of cardinal and ordinal numbers, and use the definitions to prove basic facts about arithmetical operations applied to these numbers;
b) carry out cardinal and ordinal arithmetic in specific cases and determine the cardinalities of specified sets;
d) explain how the axioms of Zermelo-Fraenkel set theory establish the existence of specified sets;
c) recognize and apply some equivalents of the axiom of choice;
d) give simple consistency and independence proofs based on the levels of the cumulative hierarchy.


1. Axiom of comprehension and Russell¿s paradox. Sets and proper classes. Simple constructions for sets and relations as sets of ordered pairs.
2. The Zermelo-Fraenkel axioms and their use to justify various constructions. Recognition of various abstraction terms as denoting sets or proper classes.
3. Ordering relations. Isomorphisms between ordered sets. Well-ordered sets.
4. von Neumann definition of ordinal numbers. Successor and limit ordinals. Proof and definition by transfinite induction.
5. The cumulative hierarchy.
6. Constructions of the standard number systems in ZF-natural numbers, integers, rationals, reals and complex numbers.
7. Axiom of choice, Zorn¿s Lemma, and the well-ordering theorem.
8. Cardinal arithmetic. Cantor¿s Theorem. Schröder-Bernstein Theorem. Cardinals as initial ordinals. Addition and multiplication. Cardinalities of various well-known sets. Generalized continuum hypothesis.

Teaching methods

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

Delivery typeNumberLength hoursStudent hours
Example Class51.005.00
Private study hours117.00
Total Contact hours33.00
Total hours (100hr per 10 credits)150.00

Private study

Reading lecture notes: 78 hours;
Solving coursework problems: 25 hours;
Preparing for examination: 14 hours.

Opportunities for Formative Feedback

Regular examples sheets.

Methods of assessment

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

Exam typeExam 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

Reading list

The reading list is available from the Library website

Last updated: 27/04/2009


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