>>14353378First of all he was a logician/philosopher and not a mathematician. He never worked in analysis, geometry or number theory (except perhaps that little encoding thing, but that's so trivial that a 5 year old could come up with it).
His most celebrated achievements are the so-called completeness and incompleteness theorems. His completeness theorem was merely a solution to an exercise in a book of Ackermann, and as a proof it wasn't that great. It was later greatly simplified by Henkin. His incompleteness theorem was also a matter of time. The ideas for the proof were already in the air, it's just that Hilbert's school was too dogmatic about the divorce of meaning from syntax as noted by Takeuti. If not Godel, other young people would've discovered the theorem pretty soon.
His program of studying large cardinal axioms to settle CH turned out a complete failure. Set theorists cope by saying it's an interesting study on its own, but it's primary aim as proposed by Godel has unquestionably failed.
Mathematicians need to start accepting that CH is not a definite problem, that set theory is superstition and start being rigorous in their thinking.
