On the Philosophical Relevance of Godel's Incompleteness Theorems

Godel began his 1951 Gibbs Lecture by stating: Research in the founda tions of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics. (Godel 1951) Godel is referring here especially to his own incompleteness theorems (Godel 1931). Godel's first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arith metic, there exists a sentence GF of the language of the system which is true but unprovable in that system. Godel's second incompleteness theo rem states that no consistent formal system can prove its own consisten cy.1 These results are unquestionably among the most philosophically important logico-mathematical discoveries ever made. However, there is also ample misunderstanding and confusion surrounding them. The aim of this paper is to review and evaluate various philosophical interpreta tions of Godel's theorems and their consequences, as well as to clarify some confusions.