I am indeed curious as to why so many researchers latch on to undecidability as a problem that relates to general intelligence? Perhaps you may have a good explanation.
I am not a mathematician, so I cannot say how much mathematics is affected in practice. Mathematicians seem good at routing around the non computable issues and using numerical approximations where they have to. Physics may be more affected where they don't realize the problem.
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I can't recall where proof by contradiction is used in Physics proofs. Usually we just cut out small numbers or execute infinity minus infinity (i.e. renormalization).
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