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Dedekind's Subtle Knife
“Think about the knife tip. That is where you are. Now feel with it, very gently. You’re looking for a gap so small you could never see it with your eyes, but the knife tip will find it, if you put…
We don’t have to postulate the existence of the cube root of 2; we actually construct a surrogate for it in the form of a cut, a way of splitting the rational numbers into two sets. You aren’t alone in wishing to do this; in the past century some mathematicians have given constructions of the real number systems that use decimal expansions as their starting point. I agree that there is something troublingly obscure about the decimal expansions of irrational numbers when you consider that for instance no intelligent being in this universe is likely to ever know the googolplexeth digit of π.
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