# MSc Pure Mathematics

Year of entry: 2024

## Course unit details:Set Theory

Unit code MATH63022 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 2 No

### Overview

The study of abstract set theory was started by Georg Cantor who, whilst studying trigonometric series, came up against problems involving iterative processes which could be applied more than a finite number of times. Out of this work came the revolutionary idea of transfinite numbers, which could be used to compare the sizes of infinite sets. A naive approach to set theory leads to paradox and it was left to Zermelo to propose an axiomatic approach that puts set theory on a sound rigorous basis.

We will study Zermelo-Fraenkel axioms for set theory, and redo some of the material from the Mathematical Logic course in this formal setting.  We will also look at the role of the Axiom of Choice, in both set theory and other parts of the mathematics.

We will then study cardinal arithmetic in some detail before moving on to some infinite combinatorics. The combinatorics will be applied to prove Silver's theorem, that the generalised continuum hypothesis cannot first fail at singular cardinals of uncountable cofinality.

### Pre/co-requisites

Students are not permitted to take MATH43021 and MATH63021 for credit in an undergraduate programme and then a postgraduate programme.

### Aims

To introduce students to set theory and its role in mathematics.

### Learning outcomes

On completion of this course students will be familiar with:

• Formalise the basic concepts of set theory in the system ZF.
• Apply the axioms of ZF to construct ordinals and cardinals.
• Formulate and proof basic properties of cardinality in ZF.
• Distinguish those arguments which require the axiom of choice.
• Prove equivalences between various forms of the axiom of choice.
• Prove various inequalities in cardinal arithmetic.
• Apply the generalised continuum hypothesis to determine cardinal exponentials.
• Prove basic results on clubs and stationary sets.

### Syllabus

2. Well-orderings, ordinals and transfinite induction [6]
3. The size of sets [4]
4. The axiom of choice [3]
5. Cardinal arithmetic [5]
6. Clubs, stationary sets, and Silver's theorem.  [5]

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• One coursework assignment: weighting 20% each,
• End of semester examination:  weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

There is no recommended textbook for this course but the following text books cover much of the material.

1. H.B. Enderton, elements of Set Theory, Academic Press.
2. K. Ciesielski, Set Theory for the Working Mathematician, London Mathematical Society Student Texts.
3. K. Hrbacek, T. Jech, Introduction to Set Theory, CHapman & Hall/CRC Pure and Applied Mathematics.
4. Y.N. Moschovakis, Notes on Set Theory, Springer-Verlag Undergraduate Texts in Mathematics.
5. K. Kunen, Foundation of Mathematics, College Publications.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 126

### Teaching staff

Staff member Role
Gareth Jones Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and t