Gödel's incompleteness theorems show that arithmetic is either inconsistent or incomplete, and that arithmetic cannot prove its own consistency. It is useful to believe that arithmetic is consistent, and therefore also incomplete, but there are other points of view.
It seems to me that both views are compatible with the consistency of first-order logic itself. Thus, I am wondering: What is the autopsy of the efforts to revisit logic in the wake of Gödel? Is there any possible world in which 2+2=5, even if the shortest proof of it is gigabytes long, and individually we have no time to verify it without use of a computer? How do contemporary philosophers interpret Gödel as applied to this specific issue?