# 2008/09 Undergraduate Module Catalogue

## MATH3102 Mathematical Logic 2

### 15 creditsClass Size: 100

**Module manager:** Professor S. B. Cooper**Email:** S.B.Cooper@leeds.ac.uk

**Taught:** Semester 2 View Timetable

**Year running** 2008/09

### Pre-requisites

MATH2040 | Mathematical Logic 1 |

### This module is mutually exclusive with

MATH5102M | Advanced Logic |

**This module is 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? (answer: no) Are mathematicians necessary? (answer: yes). The main goal is to prove Godel'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.### Objectives

To develop fundamental concepts and techniques of Mathematical Logic sufficient to prove Godel's Incompleteness Theorems, and to relate them to notions of computability, decidability and undecidability. To enable students to understand and write formal proofs in logical style. 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 Incompleteness 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.

### 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.

### Teaching methods

Delivery type | Number | Length hours | Student hours |

Example Class | 7 | 1.00 | 7.00 |

Lecture | 26 | 1.00 | 26.00 |

Private study hours | 117.00 | ||

Total Contact hours | 33.00 | ||

Total hours (100hr per 10 credits) | 150.00 |

### 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: 24/07/2009

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