Past exam of the logic and set theory course of the University of Oxford / 2010 / A / 3 / c / ii

The downward Löwenheim-Skolem theorem says that a structure in a countable language has a countable elementary substructure containing any specified finite or countable set of its elements. More generally the size bound is $\max (|\mathcal{L}|,|C|,{\aleph }_0)$ for a required parameter set $C$. In particular any satisfiable theory in a countable language has a model of size at most countable.

To see the last assertion directly from the completeness construction, add only countably many witness constants in the Henkin construction. There are then countably many finite closed terms, and taking their quotient cannot increase cardinality. The resulting term model is therefore countable. For the elementary-substructure version, perform the witness construction inside the given structure, choosing actual witnesses and closing under the function symbols. The induced substructure is elementary by the Tarski-Vaught test. This is the same witness mechanism as the completeness construction, with the ambient structure supplying the chosen witnesses.

The upward companion states that a theory with an infinite model has models in every sufficiently large infinite cardinal, at least $|\mathcal{L}|+{\aleph }_0$. Add that many pairwise unequal constants, apply compactness, and then the size-bounded construction. The countable consequence is the part needed below.