On Gödel’s Incompleteness Theorem

Abstract

Godel's Incompleteness Theorem is about the logic of mathematics. It is that a certain mathematical structure is so rich that its theory cannot be completely axiomatized. This means there will always be true statements about the structure that cannot be proved as theorems from previously given axioms. To give meaning to this conclusion, we review some examples of mathematical theorems, and their proofs, in geometry, algebra, and logic; we also give an example of a structure that is so simple (while still being interesting) that its theory can be completely axiomatized. First we look at a couple of popular descriptions of Godel's Theorem; these can be misleading. We pass to Raymond Smullyan's interpretation of Godel's Theorem as a puzzle; then to an analogy with the incompleteness of an English guide to English style. Godel's argument relies on converting statements about numbers into numbers themselves; we note how to argue similarly by understanding geometrical statements as geometrical diagrams. Geometry is thus somehow incomplete; likewise, physics