Kurt Gödel, Paper on the Incompleteness Theorems (1931)

G��del's incompleteness results are two of the most fundamental and important contributuions to logic and the foundations of mathematics. G��del showed that no axiomatizable formal system strong enough to capture elementary number theory can prove every true sentence in its language. This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, eg, the theory of computability.

Science, 2002

Revue Internationale de Philosophie, 2005

From the blurb: "In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book - extensively rewritten for its second edition - will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathem...

People normally believe that Arithmetic is not complete because Gödel launched that idea a long time ago, and nobody seems to have presented sound evidence on the contrary. We here intend to do that perhaps for the first time in history. We prove that what Stanford Encyclopedia has referred to as Theorem 3 cannot be true, and, therefore, if nothing else is presented in favour of Gödel's thesis, we actually do not have evidence on the incompleteness of Arithmetic: All available evidence seems to point at the extremely opposite direction.

This article raises some important points about logic, e.g., mathematical logic.

2020