MATH336-16S2 (C) Semester Two 2016

# Foundations of Mathematics

 15 points, 0.1250 EFTS11 Jul 2016 - 13 Nov 2016

## Description

An introduction to the philosophy of mathematics, classical and intuitionistic logic, set theory, and Gödel's theorems.

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

This course will provide students with an opportunity to develop the Graduate Attributes specified below:

 Critically competent in a core academic discipline of their award Students know and can critically evaluate and, where applicable, apply this knowledge to topics/issues within their majoring subject.

## Pre-requisites

30 points in MATH or EMTH at 200 level, as approved by the Head of School.

## Restrictions

MATH 208, MATH 308

## Assessment

Assessment Due Date Percentage
Take-Home Test 1 30%
Take-Home Test 2 30%
Take-Home Test 3 40%

## Indicative Fees

Domestic fee \$720.00

International fee \$3,450.00

* Fees include New Zealand GST and do not include any programme level discount or additional course related expenses.

For further information see Mathematics and Statistics.

## All MATH336 Occurrences

• MATH336-16S2 (C) Semester Two 2016