Full Idea
The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model.
Gist of Idea
Löwenheim-Skolem says if the sentences are countable, so is the model
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Reference
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.281
A Reaction
There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory.