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General


This course is part of the MAGIC core.

Description

This course is an introduction to set theory, focusing on foundational issues but with an eye also on the study of combinatorial properties of infinite objects.
We will start by motivating and introducing ZFC. Then we will develop the basic theory of the ordinals and cardinals in this theory, and will prove some classical theorems of combinatorial flavour. Possible topics may include cardinal arithmetic, Aronszajn trees, infinite Ramsey theory and/or some results on determinacy of games. Time permitting, I will briefly discuss large cardinal axioms, the independence phenomenon, and the problem of finding natural extensions of ZFC.
One of the goals of the course is to engage a working mathematician into looking at the foundations of the mathematical building.

Semester

Autumn 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Wed 14:05 - 14:55

Prerequisites

There are no prerequisites for this course, except for a reasonable level of mathematical maturity. Having been exposed to a course in mathematical logic would be desirable but not necessary. In fact I will give brief introductions to the relevant notions from logic.

Syllabus

Naive set theory: Sets as foundational framework for mathematics. Paradoxes.
Axiomatic set theory: ZFC.
Ordinals and cardinals. Transfinite recursion and induction. The cumulative hierarchy.
Countable and uncountable sets.
The Axiom of Choice.
Basic cardinal artihmetic.
Some combinatorial set theory: Aronszajn trees, infinite Ramsey theory.
Determinacy of infinite games.
Large cardinal axioms: Weakly compact, measurable, and beyond.
Natural axioms for mathematics: Extending ZFC.

Lecturer


David Aspero
Email d.aspero@uea.ac.uk
Phone +44 (0)1603 591433
vcard
Photo of David Aspero
Profile: I am a lecturer in Pure Mathematics at the School of Mathematics of the University of East Anglia. My work in mathematics is in set theory, and more specifically in infinite combinatorics, forcing, forcing axioms, large cadinals, definability issues, and the interactions between these areas.


Students


Photo of Hussien Abugirda
Hussien Abugirda
(Reading)
Photo of Birzhan Ayanbayev
Birzhan Ayanbayev
(Reading)
Photo of Robert Bickerton
Robert Bickerton
(Newcastle)
Photo of Gabriele Bocca
Gabriele Bocca
(East Anglia)
Photo of Philip Carter
Philip Carter
(Liverpool)
Photo of Junbin Chen
Junbin Chen
(Durham)
Photo of Lorenzo De Biase
Lorenzo De Biase
(Cardiff)
Photo of Alberto ESPUNY DIAZ
Alberto ESPUNY DIAZ
(Birmingham)
Photo of Zoltan Kocsis
Zoltan Kocsis
(Manchester)
Photo of Konrad Krolicki
Konrad Krolicki
(Lancaster)
Photo of Xiao Ma
Xiao Ma
(Loughborough)
Photo of Joel Mitchell
Joel Mitchell
(Birmingham)
Photo of Entesar Nasr
Entesar Nasr
(Liverpool)
Photo of Matty Van Son
Matty Van Son
(Liverpool)


Bibliography


Set theory: an introduction to independence proofsKenneth Kunen
Set Theory: The Third Millenium Edition, Revised and ExpandedThomas Jech
A mathematical introduction to logic (2nd. edition)Herbert Enderton


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



The assessment for this course will be via a take-home exam to be done over the exam period, to be marked as pass/fail. The paper will be made available just before the start of the exam period. The paper will comprise 6 questions. In order to pass it will suffice to obtain 50 marks out of 100.

MAGIC Set theory exam paper

Files:Exam paper
Released: Monday 26 December 2016 (58.6 days ago)
Deadline: Sunday 22 January 2017 (30.6 days ago)
Instructions:

This paper will be marked as pass/fail. Attempt FIVE out of the following six question. Do not submit solutions to all six questions. In order to pass you are required to obtain 50 marks out of 100. With the exceptions of the results you are asked to prove, you may use any of the results we have seen in the lectures. However, you must always clearly quote the theorems you are using. You may consult any references you want.

You may send me your answers as PDF's (e.g., PDF's of scans of handwritten answers) before the deadline, whcih I will then mark.



Recorded Lectures


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