Aims: Set theoretical concepts and formulations are pervasive in modern mathematics. For this reason it is often said that set theory provides a foundation for mathematics. Here 'foundation' can have multiple meanings. On a practical level, set theoretical language is a highly useful tool for the definition and construction of mathematical objects. On a more theoretical level, the very notion of a foundation has definite philosophical overtones, in connection with the reducibility of knowledge to agreed first principles.

Content:

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Overview of MA3H3 Set Theory with attention to the formulation of the ZFC axioms and the main theorems.
Cardinal Arithmetic, with and without Axiom of Choice.
Generalized Continuum Hypothesis.
Applications of Replacement.
Transfinite Recursion.
Large Cardinals.
Grothendieck Universes.

Objectives: By the end of the module, students should be able to:

• Formally state the axioms of Zermelo-Fraenkel set theory.
• Rigourously compare sizes and orderings of sets by means of explicit constructions of injections and bijections, and give interpretations in the terminology of cardinal and ordinal arithmetic.
• Outline the construction of the real number system, though various stages, ultimately from first principles.
• Give examples of mathematical statements which are equivalent to the Axiom of Choice, notice the use of this principle in mathematical arguments, and avoid unnecessary use of it.
• Appreciate the strengths, and also some of the shortcomings, of Zermelo-Fraenkel set theory as a foundation for mathematics.

Books:

• Introduction to Set Theory, K. Hrbacek and T. Jech
• Set Theory, T. Jech (a comprehensive advanced text which goes well beyond the above syllabus)
• The Axiom of Choice, T. Jech