Some different versions of completeness of a formal system, of a logic and of a theory

Solution 1:

As the quotations reveal, there are indeed two different notions here.

1 & 2 belong together. A formal theory $T$ is negation-complete if for any sentence of $T$'s language, either $T$ proves $\varphi$ or $T$ proves $\neg\varphi$ (in symbols, either $T \vdash \varphi$, then $T \vdash \neg\varphi$).

3 & 4 belong together. A logical system (e.g. for quantificational logic) is complete if whenever the premisses $\Gamma$ semantically entail the conclusion $\varphi$, there is a formal deduction in the system of $\varphi$ from premisses in $\Gamma$. (In symbols, if whenever $\Gamma \vDash \varphi$, then $\Gamma \vdash \varphi$.)

So quite different ideas: one is to do with how a deductive logic relates to the relevant notion of semantic entailment; the other is a purely syntactic notion. You can have negation-incomplete theories with complete logics, and negation-complete theories with incomplete logics (as well as the other two combinations).

Famously Gödel proved that essentially the Hilbert/Ackermann axiomatic version of quantificational logic is complete -- every semantically valid quantificational consequence (assuming a classical semantics) can be formally derived in that axiomatic system. Hence Gödel's completeness theorem (about a system of logic).

Famously Gödel proved that essentially a cut-down version of Principia's formal theory is not negation-complete: there are sentences which can't be decided one way or the other by the axioms. And the proof generalizes to any consistent formal theory that can "do enough arithmetic" (and which satisfies another condition which doesn't matter here). Hence Gödel's incompleteness theorem (about theories containing enough arithmetic).

Two different Gödelian results about different notions (so, there's no tension between them!).