Gödel's first incompleteness theorem

Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I.

The formal theorem is written in highly technical language. The broadly accepted natural language statement of the theorem is:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,^[1] but not provable in the theory (Kleene 1967, p. 250).