Full Idea
Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ.
Gist of Idea
If a first-order theory entails a sentence, there is a finite subset of the theory which entails it
Source
Wilfrid Hodges (First-Order Logic [2001], 1.10)
Book Reference
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.29
A Reaction
If entailment is possible, it can be done finitely.