David Peat

AppendixGÖDEL’S THEOREMEarlier attempts to demonstratethe consistency or the completeness of mathematics used a system ofsymbolic logic to make statements about theorems and mathematicalarguments. In other words, the status of mathematics was being examinedand certified by a system that lay outside itself—by metamathematics.Gödel’s approach was to develop a system that could makestatements about itself. The system that talked about mathematicswould itself be a part of mathematics, rather than lying outside themathematics it discussed.But how can a symbolic system be made to refer to itself? And howcan metamathematics be incorporated into mathematics? Gödel’s answerwas ingenious in the extreme. His first step was to show that toevery statement in metamathematics there corresponds a unique number.Since numbers are always part of mathematics then statements inmetamathematics about theorems and their proofs can now be reducedto the manipulation of numbers in mathematics.217