MATH432-12S2 (C) Semester Two 2012

Foundations of Mathematics

0.1250 EFTS
09 Jul 2012 - 11 Nov 2012


Foundations of Mathematics

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
    • University Graduate Attributes

      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.


Subject to approval of the Head of School.

Streams Day Time Where Notes
Stream 01 Tuesday , Thursday 8:00am-9:00am Erskine 445 9 Jul - 19 Aug,
3 Sep - 14 Oct
Wednesday 1:00pm-2:00pm Erskine 445 9 Jul - 19 Aug,
3 Sep - 14 Oct

Course Coordinator / Lecturer

Douglas Bridges


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

Indicative Fees

Domestic fee $788.00

* 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.

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