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2012/13 Taught Postgraduate Module Catalogue
MATH5103M Advanced Logic
20 creditsClass Size: 20
Module manager: Dr Andy Lewis
Email: andyl@maths.leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2012/13
Pre-requisites
MATH2040 | Mathematical Logic 1 |
This module is mutually exclusive with
MATH3102 | Mathematical Logic 2 |
Module replaces
MATH5102MThis module is not approved as an Elective
Module summary
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.Objectives
On completion of this module, students should be able to:a) carry out elementary proofs in first-order formal logic and Peano Arithmetic;
b) prove representability and recursiveness of basic number-theoretic functions and relations;
c) understand and reproduce proofs of Godel's Completeness Theorem for predicate logic, Godel's Incompleteness Theorems for Peano Arithmetic, Lob's Theorem and related results;
d) describe connections between incompleteness, consistency, computability and undecidability;
e) show a capacity for independent study, including further development of the theory via a range of more challenging homework problems, and through an oral presentation.
Syllabus
- Revision of first-order logic including Godel's Completeness Theorem
- the axiomatic method and formal Peano Arithmetic
- recursive functions and representability
- the arithmetization of syntax and Godel's First Incompleteness Theorems
- Lob's Theorem and the Second Incompleteness Theorem
- consistency, undecidability and computability
- non-standard models of arithmetic.
Additional topics chosen from: decidable fragments of arithmetic, Rosser's theorem, reflection principles, creative sets and Turing computability of theories.
Teaching methods
Delivery type | Number | Length hours | Student hours |
Lecture | 44 | 1.00 | 44.00 |
Private study hours | 156.00 | ||
Total Contact hours | 44.00 | ||
Total hours (100hr per 10 credits) | 200.00 |
Private study
- Reading lecture notes- Solving coursework problems
- Preparing for examination and advanced reading.
Opportunities for Formative Feedback
Regular problems sheets, supplemented by class discussions and two office hours per week.Methods of assessment
Exams
Exam type | Exam duration | % of formal assessment |
Standard exam (closed essays, MCQs etc) | 3 hr | 100.00 |
Total percentage (Assessment Exams) | 100.00 |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
Reading list
The reading list is available from the Library websiteLast updated: 08/01/2013
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