Editing Gödel's incompleteness theorems (section)

=== Extensions of Gödel's original result ===

Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.

Gödel demonstrated the incompleteness of the system of ''[[Principia Mathematica]]'', a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness.  In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.

Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but ''[[omega-consistent|ω-consistent]]''. A system is '''ω-consistent''' if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate {{mvar|P}} such that for every specific natural number {{mvar|m}} the system proves {{math|~''P''(''m'')}}, and yet the system also proves that there exists a natural number {{mvar|n}} such that {{mvar|P}}({{mvar|n}}). That is, the system says that a number with property {{mvar|P}} exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. {{harvard citations |txt=yes |first=J. Barkley |last=Rosser |author1-link=J. Barkley Rosser |year=1936}} strengthened the incompleteness theorem by finding a variation of the proof ([[Rosser's trick]]) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.