How do we, in general, know, verify, or otherwise prove (in meta or in FOL) an alleged infinity exists?

Many outstanding difficult mathematical problems (e.g., abc conjecture, Twin Prime conjecture, “There are infinitely many counter examples of GoldBach Conjecture”, …) would involve some set in which its being infinite or otherwise has been only alleged – not proved. It seems natural then to ask if there’s a generic way to prove (in meta or in FOL) an alleged infinity exists. (Thus the question).