Full Idea
It is strange that compactness is often ignored in discussions of philosophy of logic, since the most important theories have infinitely many axioms.
Gist of Idea
Compactness is important for major theories which have infinitely many axioms
Source
Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
Book Reference
'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.38
A Reaction
An example of infinite axioms is the induction schema in first-order Peano Arithmetic.