The incompleteness theorems

Abstract

This thesis presents a proof of the infamous First Incompleteness Theorem, which roughly states that there is no adequate axiomatisation for describing the natural numbers which can be used to uncover all the truths about them. A proof for the Second Incompleteness Theorem is also given, which states that no strong enough axiomatisation of the natural numbers is capable of proving its own consistency. The material is presented such that no formal background in mathematical logic is necessary for the reader to follow. To guide the intuition, a brief contextual explanation of what we shall be doing is given in the introduction. After presenting Unsorted First Order Logic, we explore the crucial notion of computability, arguably the most important piece of the puzzle in proving the Incompleteness Theorems.