Full Idea
From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'.
Gist of Idea
The completeness of first-order logic implies its compactness
Source
David Bostock (Philosophy of Mathematics [2009], 5.5)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.153
A Reaction
The point is that the completeness requires finite proofs.
Related Idea
Idea 10974 Compactness is when any consequence of infinite propositions is the consequence of a finite subset [Read]