thinking about points in the plane in terms of coordinates lets us reduce geometric questions to algebra. this is, broadly speaking, one of the most successful mathematical strategies ever, and mathematicians have taken it very far
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this would not work if, for example, we tried to use the rational numbers instead of the real numbers. circles with only rational coordinates would be "missing" a lot of points (with irrational coordinates) and that would produce the wrong answers to various questions
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so the real numbers "fill in gaps" that are "missing" in the rational numbers, and once you've filled those gaps the real numbers can implement euclidean geometry
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so, okay, that's pretty good, why does anyone have a problem with the real numbers then (and they do)? the problem is that we pay a very bizarre price for filling in the gaps: almost every specific real number is literally indescribable, because there are too many of them!
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the problem is that no matter how you choose to describe real numbers, there are only countably many possible descriptions (e.g. only countably many programs that spit out strings of digits), but uncountably many real numbers! it's wack
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this makes some people very uncomfortable (and i think that discomfort is justified). what is "real" about the vast majority of the real number line being inaccessible to any form of description whatsoever???
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(and i mean *vast* majority - in a precise technical sense the probability of a randomly chosen real number being describable is literally zero)
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for the purposes of simpler questions in euclidean geometry you can get away with working with a much smaller set of numbers, the algebraic reals, which are all describable
but you actually need all of the real numbers to do calculus. and we need calculus for a million things
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so the real numbers, as usually constructed, are (this is very much in-my-opinion) this philosophically unsatisfying technical kludge we put up with because it lets us put geometry and calculus and a million other things on a rigorous foundation
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Aren't there counting arguments that whatever numbers we do use for geometry or calculus would have to be uncountable, and get the "0% are describable" property? I'm surprised you're optimistic about this.
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i have hopes that we move away from thinking of the real numbers as a set in the first place
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Maybe only Borel sets? I like real numbers but they sort of stink? Borel set of reals can do calculus I think and is less poopy (no uncountably large sets in any construction)? Am I on the right track here?
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Or are you saying you don't like even a single indescribable real number? Because I mean we have indescribable integers too if you count that Rayo number and definitely tons of sequences that aren't computable.