Introductory description

See learning outcomes.

Module web page

Module aims

Outline syllabus

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.

The module will commence with a brief review of naive set theory. Unrestricted set formation leads to various paradoxes (Russell, Cantor, Burali-Forti), thereby motivating axiomatic set theory. The Zermelo-Fraenkel system will be introduced, with attention to the precise formulation of axioms and axiom schemata, the role played by proper classes, and the cumulative hierarchy picture of the set-theoretical universe. Transfinite induction and recursion, cardinal and ordinal numbers, and the real number system will all be developed within this framework. The Axiom of Choice, and various equivalents and consequences, will be discussed, and various other principles known to be ZF-independent, such as the Continuum Hypothesis and the existence of Inaccessible Cardinals, will also be touched on.

Learning outcomes

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.

Indicative reading list

Set Theory, Jech
This is a comprehensive advanced text which goes well beyond
the above syllabus.
Notes on set theory, Y. Moschovakis
Elements of set theory, H. Enderton
Introduction to set theory, Hrbacek and Jech

Subject specific skills

Appreciate the strengths, and also some of the shortcomings, of Zermelo-Fraenkel set theory as a foundation for mathematics.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.