Full Idea
The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem.
Gist of Idea
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
Book Reference
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.172
A Reaction
On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223).