Module manager: Prof Michael Rathjen
Email: M.Rathjen@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2024/25
MATH2040 or MATH2041 or MATH2042, or equivalent
| MATH2040 | Mathematical Logic 1 |
| MATH2041 | Logic |
| MATH2042 | Logic with Computation |
| MATH5104M | Advanced Proof and Computation |
This module is not approved as an Elective
Metamathematics and proof theory try to answer fundamental questions about axiomatic theories (e.g. number theory) like: - Are they consistent (free from contradiction)? - How do we know? - Could they be developed by computers without human assistance? - Are mathematicians necessary? The main goal is to prove Gödel's Incompleteness Theorems (1931) which show that if a formal theory has strong enough axioms then there are statements which it can neither prove nor refute. This module will also provide background to the impact of Gödel's Theorem on the modern world, and the way it sets an agenda for further research.
On completion of this module, students should be able to:
-carry out elementary proofs in systems of arithmetic;
- prove representability and recursiveness of basic number-theoretic functions and relations;
- understand and reproduce proofs of Gödel's Incompleteness Theorem and related results;
- describe connections between incompleteness, consistency, computability and undecidability;
- show a capacity for independent thinking, including further development of the theory via a range of more challenging homework problems.
- An understanding of basic systems of arithmetic;
- an understanding of the notions of compubtability, recursive enumerability and representability;
- A knowledge of Gödel's Incompleteness Theorem, its proof and implications.
Diagonalisation
Understanding how a basic computing device works and can be programmed
- Revision of first-order logic including Gödel's Completeness Theorem;
- the axiomatic method and basic systems of arithmetic;
- register machines;
- recursive functions and representability;
- the arithmetization of syntax and Gödel's First Incompleteness Theorem;
- consistency, undecidability and computability.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
Homework questions in which the student is asked to display independent thinking in order to further develop the theory described in class.
Example classes and final exam.
| Exam type | Exam duration | % of formal assessment |
|---|---|---|
| Open Book exam | 2.0 Hrs 30 Mins | 100 |
| Total percentage (Assessment Exams) | 100 | |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
There is no reading list for this module
Last updated: 2/18/2025
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