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To comment on your philosophical proposal, Greg. I don't know very much about the Riemann Hypothesis, (say), but is it important enough to be an axiom? For example, Fermat's Last Theorem was of little significance to number theory (so I've read), by comparison with the mathematical discoveries made in the attempt to prove it. How should a mathematician decide when enough is enough, and consign an otherwise useless hypothesis to the axiomatic waste bin?