Set Theory

MATH43022 Set Theory 15 Semester 2 Undergraduate Level 4 Gareth Jones Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 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. 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. To introduce students to set theory and its role and use in mathematics. On completion of this course students will be familiar with:  <ul> <li> Formulate and prove basic properties of cardinality in ZF.</li> <li> Apply the axioms of ZF to construct ordinals and cardinals.</li> <li> Formulate and proof basic properties of cardinality in ZF.</li> <li> Distinguish those arguments which require the axiom of choice.</li> <li> Prove equivalences between various forms of the axiom of choice.</li> <li> Prove various inequalities in cardinal arithmetic.</li> <li> Apply various hypotheses to determine cardinal exponentials in certain cases.</li> <li> Prove basic results on clubs and stationary sets.</li> </ul>   <ol> <li> Paradoxes and axioms [4]</li> <li> Well-orderings, ordinals and transfinite induction [6]</li> <li> The size of sets [4]</li> <li> The axiom of choice [3]</li> <li> Cardinal arithmetic [5]</li> <li> Clubs, stationary sets, and Silver's theorem.  [5]</li> </ol>   0 Other 20% 1 Written exam 80% One coursework assessment; weighting 20% eachEnd of semester examination: weighting 80% 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.   MATH20201 Algebraic Structures 1 Pre-Requisite Compulsory MATH33011 Mathematical Logic Pre-Requisite Compulsory Please noteStudents are not permitted to take MATH43021 and MATH63021 for credit in an undergraduate programme and then a postgraduate programme. Y There is no recommended textbook for this course but the following text books cover much of the material. <ol> <li> H.B. Enderton, elements of Set Theory, Academic Press.</li> <li> K. Ciesielski, Set Theory for the Working Mathematician, London Mathematical Society Student Texts.</li> <li> K. Hrbacek, T. Jech, Introduction to Set Theory, Chapman & Hall/CRC Pure and Applied Mathematics.</li> <li> Y.N. Moschovakis, Notes on Set Theory, Springer-Verlag Undergraduate Texts in Mathematics.</li> <li> K. Kunen, Foundation of Mathematics, College Publications.</li> </ol> Lectures 26 Tutorials 13 0 111 This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates