module specification

The module is designed to be accessible to both mathematics and non-mathematics students alike. The prerequisite for this module is basic arithmetic and desire to think about abstract ideas.

This module is centred around Cantor’s theory of infinite sets. The historical background of the idea of infinity will be given from the ancient Greek philosophers up to Immanuel Kant. The main ideas behind the Cantor’s theory of transfinite numbers will be developed and then we will look at some of the consequences of Cantor’s work present in Mathematics, Computer Science and Philosophy.
 

This module aims to provide an opportunity for non-mathematics and mathematics students to study Cantor's theory of transfinite ordinal numbers and their arithmetic. Cantor's famous proof of the uncountability of the real continuum by a diagonal argument, and his revolutionary discovery that there were different "orders of infinity" - indeed infinitely many such - will feature prominently in our basic study of infinite cardinal numbers and their arithmetic. We shall see how the pitfalls of the various early "set theoretic paradoxes" such as that of Zeno (Achilles and tortoise) and Russell ("the set of all sets that do not contain themselves") were avoided.

The idea of infinity through history: Greeks to Kant. Cantor and the origin of his Set Theory. Transfinite numbers: Ordinals (definition and arithmetic). Transfinite numbers: Cardinals (definition and arithmetic). Cantor's Theorem and Continuum Hypothesis. Self-reference and Godel's Incompleteness Theorem(s). Limitations of Thought.

The module will be taught in 3-hour blocks divided into 1-hour lecture and 2-hour tutorial. Tutorials will be used for open discussion of the main topics covered in lectures. Students will be asked to read relevant material beforehand and offer their understanding during the tutorials. In addition to standard VLE presence there will be links available for further readings and discussion groups. 

LO1: Understand the origins of Cantor's set theory.
LO2: Understand definitions and arithmetic of transfinite numbers.
LO3: Understand the implications of the limitations of any formal system.
 

There will be a comprehension test in form of a take-away work which will give students opportunity to demonstrate their understanding of selection of topics(LO1, LO2). The final assessment will be an essay where students will select one question and write an in-depth critical analysis (LO1 to LO3).

Dauben, J. W.( 1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, ISBN 0-691-02447-2.
Rucker, R.( 1995), Infinity and the Mind, Princeton University Press,  ISBN 0-69-100172-3
Priest, G. (2006), Beyond the Limits of Thought, Oxford University Press, ISBN 0-19-924421-9.
Fatconi, T.G.( 2006), The Mathematics of Infinity, John Wiley & Sons, Inc., ISBN 10 0-471-79432-5.