MATH432-12S2 (C) Semester Two 2012
Foundations of Mathematics

Description

What is mathematics? What is the connection between truth and provability? What are the limitations of mathematics? Is every mathematical truth provable, at least in principle? "Foundations of mathematics" is the name given to the examination of such questions using mathematical methods wherever applicable. The technical mathematical background required for this course is fairly minimal (for example, it does not require a background course on analysis), so the course should be accessible to some philosophers and computer scientists; nevertheless, a measure of mathematical/intellectual maturity is an advised prerequisite.

Part 1: Classical propositional and predicate logic; valuations, proof trees, and derivations; soundness and completeness; the compactness theorem and its applications

Part 2: Axiomatic Zermelo-Fraenkel set theory; ordinals; the axiom of choice and the continuum hypothesis; independence of axioms.

Part 3: Gödel's incompleteness theorems.

Learning Outcomes

By the end of the course, the students will have an appreciation of some of the major breakthroughs in mathematical logic in the first 70 years of the twentieth century, and they will be able to carry out technical work in formal logic and set theory.

In particular they should:

• understand the fundamentals of proof and model theory, for classical propositional and predicate logic, up to the Gödel-Henkin completeness theorem, the compactness theorem, and the upward Löwenheim-Skolem theorem
• understand, and use, the axioms of Zermelo-Fraenkel set theory, in particular in the development of the theory of ordinals
• understand the proof and the significance of Gödel's incompleteness theorems

Pre-requisites

For further information see Mathematics and Statistics.