Full Idea
Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures.
Gist of Idea
Model theory uses sets to show that mathematical deduction fits mathematical truth
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
Book Reference
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.-8
A Reaction
That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them.