Full Idea
First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis.
Gist of Idea
First-order logic only has its main theorems because it is so weak
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.411-2)
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.411
A Reaction
He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1).