In T. Skolem 1922 the author publishes a weak version of the Skolem-Löwenheim theorem which we call WLS and which according to Wikipedia says that every countable theory which is satisfiable in a model is also satisfiable in a countable model. My understanding is that WLS entails completeness in the sense of K. Gödel 1929. Does completeness entail WLS?

Edit:

Noah Schweber and Emil Jeřábek are right in that WLS does not literally entail completeness, nor vice versa. Nevertheless, earlier today in Equivalence between Lowenheim-Skolem Theorem and Godel Completeness I came across the following statement: “This explains why Godel (Coll. Wrks. Vol 1, p. 52) writes that Skolem's weak theorem implies completeness: "Skolem...could justly claim...that, in his 1922 paper, he implicitly proved: 'Either A is provable or ¬A is satisfiable” (“provable” taken in an informal sense).'” Clearly, 'implicitly proved' is here understood e.g. so so that information from the proof sufficed to give the completeness proof.

When it concerns the formulation of my question, I was also confounded by the title of the page I just linked to.

Memory and Google reveal that there are several authors who comment upon Gödel's statement quoted above. See e.g. pp. 50-51 in Paul Mancosu, The adventure of Reason: ...

Presumably the "implicit" connections between WLS and completeness has connections with WKL.

References:

K. Gödel 1929: *Über die Vollständigkeit des Logikkalküls*, Doctoral dissertation, University Of Vienna

Skolem 1922: *Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre*, 5th Scand. Math. Congress

statementdoesn't - we can have logics with the weak Lowenheim-Skolem property which don't have any good proof system (note that nothing in your statement of wLS has to do with proofs, and in particular you've referred tosatisfiabilityinstead of *consistency). $\endgroup$proofsof completeness of classical first-order logic also give Löwenheim–Skolem for free. $\endgroup$1more comment