In my opinion, the Wikipedia article on complete theory (mathematical logic) contains a mistake, or at least is ambiguous and misleading.

The first paragraph is fine, it gives the correct definition of complete theory in mathematical logic.

The second paragraph is problematic. It says:

This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. This theorem states that recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete.

I agree with the first two sentences. The last sentence is wrong (or at least misleading) because it attributes the content of Gödel's first incompleteness theorem to Gödel's completeness theorem, in this way not only the information is mistaken but it also confuses two different meanings of completeness.

As far as I know, in mathematical logic there are two different senses for "completeness", one for theories (for instance, Presburger arithmetic is complete, Peano arithmetic is not) and one for logics (for instance, first-order logic is complete, second-order logic is not). Gödel's completeness theorem is about completeness of first-order logic (completeness of a logic), Gödel's first incompleteness theorem is about (in)completeness of Peano arithmetic (completeness of a theory). Clearly, there is no contradiction between the two Gödel's theorems.

Do you agree with me? Please, can someone fix the error in the Wikipedia article?

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    $\begingroup$ I agree with you. In G's incompleteness theorem the logic is "complete": it can prove all semantical consequences of the axioms. In a sense, the source of the "incompleteness" must be located into the arithmetical axioms that cannot "carve out" from the set of semantical consequences those one that are true only in the "intended" model. $\endgroup$ – Mauro ALLEGRANZA Sep 14 '17 at 13:41
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    $\begingroup$ The third sentence seems to be about Godel's incompleteness theorem rather the Godel's completeness theorem. Whoever last edited the passage must have had the two badly mixed up. $\endgroup$ – realdonaldtrump Sep 14 '17 at 13:56
  • $\begingroup$ You're right. Have you considered either fixing it yourself, or starting a talk thread about it? $\endgroup$ – Rob Arthan Sep 14 '17 at 15:28
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    $\begingroup$ IMHO, this discussion should be moved to the talk page of the Wikipedia article en.wikipedia.org/wiki/Talk:Complete_theory . The change was introduced by Bbbaat en.wikipedia.org/w/… . So we just need to undo Bbbaat's changes. $\endgroup$ – beroal Sep 14 '17 at 15:30

I think the intent was correct, actually .. it was just very, very badly phrased: it points out that the completeness talked about in the article is to be differentiated from the notion of logical completeness that Godel's completeness theorem is about. With the 'this' in the third sentence I believe the author was referring back to the notion of completeness in 'this article', which is not logical completeness but something like arithmetical (in)completeness. That is, I think the author meant the third sentence to be read as [As opposed to the notion of logical completeness, this notion of completeness ...].

But yes, very poorly phrased: I would say go ahead and edit (or revert back to an earlier version as suggested in the comments)


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