Proof sketch for Gödel's first incompleteness theorem

This article gives a sketch of a proof of Gödel's first incompleteness theorem. This theorem applies to any formal theory that satisfies certain technical hypotheses which are discussed as needed during the sketch. We will assume for the remainder of the article that a fixed theory satisfying these hypotheses has been selected. Throughout this article the word "number" refers to a natural number. The key property these numbers possess is that any... more

This article gives a sketch of a proof of Gödel's first incompleteness theorem. This theorem applies to any formal theory that satisfies certain technical hypotheses which are discussed as needed during the sketch. We will assume for the remainder of the article that a fixed theory satisfying these hypotheses has been selected. Throughout this article the word "number" refers to a natural number. The key property these numbers possess is that any natural number can be obtained by starting with the number 0 and adding 1 a finite number of times. Gödel's theorem applies to any formal theory that satisfies certain properties. Each formal theory has a signature that specifies the nonlogical symbols in the language of the theory. For simplicity, we will assume that the language of the theory consists of: This is the language of Peano arithmetic. A well formed formula is a sequence of these symbols that is formed so as to have a well-defined reading as a mathematical formula. Thus x = SS0... less