Can you give some link or citation about "reflexive"? I can't figure out what this means. It seems to me that if PA can prove every finite subset of its axioms consistent, then it can prove itself consistent: "If PA is inconsistent, then there would be a statement A and a proof of A and a proof of NOT A; but each proof only uses a finite set of axioms; the union of two finite sets is finite; so there is a finite set of axioms of PA that proves both A and NOT A; but every finite subset of PA is consistent, contradiction. Ergo, PA is consistent." Clearly, I'm misunderstanding something.

(PA can't talk about sets in general, but finite sets are fine, and "finite subset of PA" just means "finite set of propositions, all of which are in the enumeration that is PA.")