MA3H315 Set Theory
Introductory description
See learning outcomes.
Module aims
Set theoretical concepts and formulations are pervasive in modern mathematics. For this reason it is often said that set theory provides a foundation for mathematics. Here 'foundation' can have multiple meanings. On a practical level, set theoretical language is a highly useful tool for the definition and construction of mathematical objects. On a more theoretical level, the very notion of a foundation has definite philosophical overtones, in connection with the reducibility of knowledge to agreed first principles.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
The module will commence with a brief review of naive set theory. Unrestricted set formation leads to various paradoxes (Russell, Cantor, BuraliForti), thereby motivating axiomatic set theory. The ZermeloFraenkel system will be introduced, with attention to the precise formulation of axioms and axiom schemata, the role played by proper classes, and the cumulative hierarchy picture of the settheoretical universe. Transfinite induction and recursion, cardinal and ordinal numbers, and the real number system will all be developed within this framework. The Axiom of Choice, and various equivalents and consequences, will be discussed, and various other principles known to be ZFindependent, such as the Continuum Hypothesis and the existence of Inaccessible Cardinals, will also be touched on.
Learning outcomes
By the end of the module, students should be able to:
 Formally state the axioms of ZermeloFraenkel set theory.
 Rigourously compare sizes and orderings of sets by means of explicit constructions of injections and bijections, and give interpretations in the terminology of cardinal and ordinal arithmetic.
 Outline the construction of the real number system, though various stages, ultimately from first principles.
 Give examples of mathematical statements which are equivalent to the Axiom of Choice, notice the use of this principle in mathematical arguments, and avoid unnecessary use of it.
 Appreciate the strengths, and also some of the shortcomings, of ZermeloFraenkel set theory as a foundation for mathematics.
Indicative reading list
Set Theory, Jech
This is a comprehensive advanced text which goes well beyond
the above syllabus.
Notes on set theory, Y. Moschovakis
Elements of set theory, H. Enderton
Introduction to set theory, Hrbacek and Jech
Subject specific skills
Appreciate the strengths, and also some of the shortcomings, of ZermeloFraenkel set theory as a foundation for mathematics.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (20%) 
Seminars  10 sessions of 1 hour (7%) 
Private study  110 hours (73%) 
Total  150 hours 
Private study description
Office hours, private study, preparation for lectures and exams, assignment sheets  110 hours
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group B1
Weighting  Study time  

Inperson Examination  100%  

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Exam Feedback
Courses
This module is Core for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy
 Year 3 of GV17 Mathematics and Philosophy

UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 4 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 4 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 3 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Optional for:
 Year 1 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)

UCSAG4G1 Undergraduate Discrete Mathematics
 Year 3 of G4G1 Discrete Mathematics
 Year 3 of G4G1 Discrete Mathematics
 Year 3 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 4 of UCSAG4G2 Undergraduate Discrete Mathematics with Intercalated Year

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 3 of G300 Mathematics, Operational Research, Statistics and Economics
 Year 4 of G300 Mathematics, Operational Research, Statistics and Economics
This module is Core option list D for:
 Year 4 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list A for:
 Year 1 of TMAAG1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 1 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 3 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year

UMAAG100 Undergraduate Mathematics (BSc)
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G100 Mathematics
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 4 of G106 Mathematics (MMath) with Study in Europe

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 3 of GF13 Mathematics and Physics
 Year 3 of GF13 Mathematics and Physics
 Year 4 of UPXAGF14 Undergraduate Mathematics and Physics (with Intercalated Year)
 Year 4 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 4 of UMAAG101 Undergraduate Mathematics with Intercalated Year

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 3 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 4 of USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)
This module is Option list B for:
 Year 1 of TMAAG1PE Master of Advanced Study in Mathematical Sciences
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 4 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 3 of GG14 Mathematics and Statistics
 Year 3 of GG14 Mathematics and Statistics
 Year 4 of USTAGG17 Undergraduate Mathematics and Statistics (with Intercalated Year)
This module is Option list E for:

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 3 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
 Year 4 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)

USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
 Year 3 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
 Year 4 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
 Year 5 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)