In 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 1922. Does Completeness entail WLS?

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