- Gödel began his 1951 Gibbs Lecture by stating: "Research in the foundations 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." (Gödel 1951)
- Gödel is referring here especially to his own incompleteness theorems (Gödel 1931).
- Gödel's first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arithmetic, there exists a sentence GF of the language of the system which is true but unprovable in that system.
- Gödel's second incompleteness theorem states that no consistent formal system can prove its own consistency.
- 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 interpretations of Gödel's theorems and their consequences, as well as to clarify some confusions.
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