Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. This article gives a sketch of a proof of Godel'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. Godel'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. A theory is consistent if there is no formula F such that both F and its negation are provable. -consistency is a stronger property than consistency.

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