How has the mathematics community reacted? With Gödel, at first there was a lot of shock. As a student in the late 1950s and early 1960s, I would read essays by Weyl, by John von Neumann, and by other mathematicians. Gödel really disturbed them. He took away their belief in the Platonic world of ideas, the principle that mathematical truth is black or white and provides absolute certainty.Yes, of course mathematicians use axiomatic theories. They have for over two millennia. That is what mathematics is.
But now, strangely enough, the mathematics community ignores Gödel incompleteness and goes on exactly as before, in what I would call a Hilbertian spirit, or following the Bourbaki tradition, the French school inspired by Hilbert. Formal axiomatic theories are still the official religion in the mathematics community. ...
Gödel incompleteness is even unpopular among logicians. They are ambivalent. On the one hand, Gödel is the most famous logician ever. But, on the other hand, the incompleteness theorem says that logic is a failure.
Godel certainly did not prove that logic is a failure. He would have vehemently resisted interpreting his work that way.