Gödel’s Theorem

Abstract

One of the highlights of modern logic is Gödel’s incompleteness theorem. It tells us that, e.g. formal arithmetic is incomplete, i.e. there is a statement φ such that neither φ nor ¬φ. can be proved. It put an end to Hilbert’s optimistic claim that (at least) Peano’s axiom system would prove all its true statements. The proof of Gödel’s theorem makes extensive use of the theory of computable (or recursive) functions. There is a self-contained elegant presentation of the required parts of recursion theory. The basic trick is the arithmetization of formal theories, in particular arithmetic that can treat (coded versions of) statements inside its theory. The arithmetization is carried out for Gentzen’s natural deduction.