Anyway, Cantor's idea of infinity (the "transfinite numbers") aren't really the same thing as what people mean when they put ∞ as the "sum" for a divergent series like 1+2+3+4+...
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A lot of people lose the thread on what ∞ is, or what a limit is. It’s a convention, a set of instructions. You can get as large/small/close as you need to for whatever practical task you’re using the numbers for. But ∞ doesn’t actually exist as a number.
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Replying to @lawnerdbarak @arthur_affect and
And that’s why the axiom of choice shit gets so weird. ∞ isn’t real, so higher order infinities are suuuuuuper weird.
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As I understand it, hyperreal numbers, invented in the 1960s, are an attempt to do what Cantor didn't and actually treat the ∞ (infinite and infinitesimal) in calculus as a number you can do math with ("As I understand it" meaning I understand almost nothing about it)
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Replying to @arthur_affect @lawnerdbarak and
I don't know much about non-standard analysis either, but if I'm not completely mistaken here you'll have an infinity of infinite numbers in that system because, as you pointed out, e.g. 1 + x != x no matter what.
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Replying to @hllizi @lawnerdbarak and
Right, like the reason for the term "hyperreal" is they're "adjacent" to the real numbers, constructed directly from doing math with them, and instead of just going "ERROR" when you get an infinite term continuing on with the infinite term attached to real terms
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Replying to @arthur_affect @hllizi and
So there's no actual, distinct number ∞ you can use by itself, just in the course of doing an integration you might get an x + n term where x is real and n is an unbounded, non-standard term But you can't ever compare that n to any other n you get doing something else
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Replying to @arthur_affect @hllizi and
Hence you just roll with paradoxical ideas like n +1 is higher than n, as is 2n or n^2, even though n supposedly is ∞
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Replying to @arthur_affect @hllizi and
It comes from the idea of treating the infinitesimal as actually a thing, like there really is such a thing as "infinitesimally more than 4" (4+ε) and therefore you can take the reciprocal of "infinitesimally more than 4" and get "1/4+infinity"
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Replying to @arthur_affect @hllizi and
But just as the ε isn't actually 0, because 4+0 would just be 4, the "infinity" isn't actually the old school (contradictory) idea of this objective highest number that exists ∞ that is the result of 1/0
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It's just this special thing sitting next to the real number you're not allowed to touch And that's the big distinction with Cantor, whose WHOLE THING was imagining infinities that relate, specifically, to other infinities, with operations that apply specifically to them
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