Gödel’s first incompleteness theorem states that for any consistent logical system S able to express arithmetic there must exist sentences that are true in the standard interpretation of S, but not provable. Moreover, if S is omega-consistent then there exist sentences such that neither they nor their negations are provable. The second theorem states that no such system can be powerful enough to prove its own consistency.

These results, published in 1931 (‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’; trs. as ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems I’), determined the limits of purely formal methods in mathematics, and in particular marked the end of Hilbert’s programme. Additional philosophical significance attaches to the way Gödel proved his first result. This was by defining a formula P that whilst unprovable can be seen to be true, given the way it is constructed. The implied moral is that truth in some way outruns provability, at least when that is considered formally.

Gödel proceeded by encoding logical formulae as numbers, so that to a statement about provability in the metatheory there corresponded by the mapping a simple statement of elementary mathematics. The formalism of arithmetic thus contains sentences that, considered via the enumeration, express propositions of their own metamathematics. The formula P is such a one: it expresses, to one interpreting it via the enumeration, its own unprovability. Gödel’s procedure thus is an instance of a diagonal argument, and it bears a vivid analogy to examples of the semantic paradoxes. It is thus extremely important that all the methods he used are perfectly rigorous.

Gödel’s procedure and his results opened the way to a fully rigorous treatment of the notion of a computable function, and to our modern understanding of the power and limits of computation, and the possibility or otherwise of programs that test for consistency and completeness.

http://www.sciam.com/print_version.cfm?articleID=0002F4F1-B4F3-1C71-9EB7809EC588F2D7 An introduction to Gödel’s theorem

http://cscs.umich.edu/~crshalizi/notebooks/godels-theorem.html A sceptical view on the uses of Gödel’s theorem, with links to other resources