Midlands Logic Seminar

Week 1: 4 October

Double study group: Richard Kaye (Birmingham)
"Mathematics and metamathematics in PA"

This session is intended to be particularly useful to students from 4th year upwards, in mathematics or a related discipline, who wants to follow later study group sessions on arithmetic and truth. The prerequisites are first order logic including soundness completeness and compactness theorems. In an informal way I will show how a huge amount of mathematics can be done in the first order theory of the natual numbers {0,1,2,...} with addition, multiplication and the order relation (and more generally the theory known as PA consisting of some basic axioms for the nonnegative parts of discretely ordered ring together with an induction axiom scheme for the first order language).

The main trick is due to Gödel and involves coding finite sequences of integers as single numbers and decoding them in this language. Using this idea much basic number theory, combinatorics and metamathematics can be developed in PA.

The approach will be informal and model theoretic. A picture of models of PA will emerge and be emphasised. Proofs inside PA will typically be done by `working inside an arbitrary model of PA' and then appealing to the completeness theorem. The idea is to develop ones intuition and present a clear idea of what PA can do.