## MATH3102 Mathematical Logic 2

### 15 creditsClass Size: 100

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

### Pre-requisites

MATH2040 or MATH3162 or MATH3163 equivalent.

This module is approved as an Elective

### 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

This is a challenging pure mathematics module requiring commitment and a willingness to study and spend time on homework. 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. These results have had a radical impact on mathematics itself and its philosophical foundations. The topics covered are: 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 26 hours; Examples classes: 7 hours.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

One 3 hour examination at end of semester (100%).