MATH43021 - 2007/2008
- Title: Set Theory
- Unit code: MATH43021
- Credits: 15
- Prerequisites: MATH20302 Propositional Logic.
- Co-requisite units: Some familiarity with predicate logic, as studied in the course unit MATH43001 would be helpful, but is not essential.
- School responsible: Mathematics
- Members of staff responsible:
To introduce students to the elements of set theory and its role as a foundation for classical mathematics.
Brief Description of the unit
The notion of set is one of the fundamental notions of modern mathematics. Set Theory was initiated by Cantor over a hundred years ago. He developed a revolutionary theory of transfinite numbers that can be used to compare the `sizes' of possibly infinite sets. The first part of the course unit will be concerned with Cantor's theory. A naive approach to set theory leads to paradox and Zermelo initiated an axiomatic approach that puts set theory on a sound rigorous basis. Axiomatic set theory can be viewed as a foundation of mathematics in the following sense. All mathematical notions can be defined in purely set theoretical terms and their properties can be proved using only the set theoretical axioms. Also the language of set theory has played a central unifying role in modern mathematics. These topics will be examined in the second part of the course unit.
On successful completion of this course unit students will have acquired
- facility with the notions of elementary set theory;
- a sound knowledge of the basic properties of the cardinal and ordinal numbers;
- familiarity with the axiom system ZF and its role as a foundation for mathematics,
- an understanding of the axiom of choice and some of its applications.
Future topics requiring this course unit
- Part I: The size of sets [15 lectures]
Standard set notation for elementary set theory; naive and not so naive set theory; finite and countable sets; cardinal and ordinal numbers - their ordering and arithmetic.
- Part II: Axiomatic Set Theory [15 lectures]
The reduction of mathematical notions to purely set theoretic ones; the ZF axioms for pure sets; the axiom of choice and the well-ordering principle; the cumulative hierarchy.
There is no recommended textbook to cover the course. The following are some good books to consult.
- A. G. Hamilton, Numbers, Sets and Axioms, CUP Press (in paperback).
- Y. N. Moschovakis, Notes on Set Theory, Springer-Verlag Undergraduate Texts in Mathematics.
- H. B. Enderton, Elements of Set Theory, Academic Press.
Teaching and learning methods
30 lectures, 6 examples classes, and assigned reading. In addition students are expected to do at least seven hours a week of private study on this course unit.