Editing Principia Mathematica (section)

===Gödel 1930, 1931===

In 1930, [[Gödel's completeness theorem]] showed that first-order predicate logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some [[Model theory|model]] of the axioms. However, this is not the stronger sense of completeness desired for ''Principia Mathematica'', since a given system of axioms (such as those of ''Principia Mathematica'') may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

[[Gödel's incompleteness theorems]] cast unexpected light on these two related questions.

Gödel's first incompleteness theorem showed that no recursive extension of ''Principia'' could be both consistent and complete for arithmetic statements. (As mentioned above, Principia itself was already known to be incomplete for some non-arithmetic statements.) According to the theorem, within every sufficiently powerful recursive [[logical system]] (such as ''Principia''), there exists a statement ''G'' that essentially reads, "The statement ''G'' cannot be proved." Such a statement is a sort of [[Catch-22 (logic)|Catch-22]]: if ''G'' is provable, then it is false, and the system is therefore inconsistent; and if ''G'' is not provable, then it is true, and the system is therefore incomplete.

[[Gödel's second incompleteness theorem]] (1931) shows that no [[formal system]] extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the ''Principia'' system" cannot be proven in the ''Principia'' system unless there ''are'' contradictions in the system (in which case it can be proven both true and false).