Talk:Complete theory

Hello, I believe the definition of complete theory given in this page is not the best one. If I'm not mistaken, the correct definition should be the one given in this page, i.e. a complete theory need not be maximal. If you give the more restrictive definition in this page, it is no longer true that a theory is complete if and only if any two of its models are elementary equivalent. 134.58.253.57 (talk) 09:05, 26 September 2012 (UTC)

Characterisation of Completeness doesn't get Consistency
"For theories in logics which contain classical propositional logic, this is equivalent to asking that for every sentence φ in the language of the theory it contains either φ itself or its negation ¬φ." from the article seems wrong, that will capture the completeness but not consistency. There may be theories satisfying this but which are not consistent, for example the theory might contain $$\phi$$ and $$\psi$$ but not $$\phi\vee\psi$$. --Catrincm (talk) 12:39, 5 February 2015 (UTC)
 * If $$\phi$$ and $$\psi$$ are formulas then $$\phi\vee\psi$$ must be a formula in the language. Wqwt (talk) 08:25, 18 February 2020 (UTC)

"negation complete"
I know that this type of completeness is sometimes called 'negation-complete' to distinguish it from the other type of completeness, as in Godel's theorems (and the article does spell this distinction out). The term 'negation complete' is used over at the article Completeness (logic) and I've linked that back here; what I _cannot_ find is an adequate reference that witnesses this usage. 71.139.124.132 (talk) 20:38, 21 November 2021 (UTC)

Is the notion of completeness only defined for consistent theories?
Forgive me if this is a trivial question (I am by no means an expert in logic), but I am unsure of the relationship between completeness and consistency. This article defines a theory to be complete if it can prove any statement or its negation, but it is not made clear whether this "or" is an exclusive-or or an inclusive-or.

My guess would be that we don't define completeness only for consistent theories, for the reason that determining whether theories are consistent is very difficult (e.g. we can't state outright that standard theories such as PA or ZF are consistent; we only have conditional statements such as "ZF is consistent => PA is consistent" and "some-large-cardinal-axiom-that-is-way-over-my-head => ZF is consistent") – however, as I say, I am not an expert so this guess could well be wrong. It would be appreciated if somebody in the know could clarify whether the aforementioned "or" is exclusive or inclusive, and perhaps also add a sentence or parenthetical about consistency in the context of complete theories. Thanks.