This is a fun one, because of course the kneejerk rejection of the statement "1+2+3+4+5+... = -1/12" is correct Under everyday, ordinary grade-school arithmetic, the answer can't be "-1/12" -- but the answer can't be any other number either, the operation itself is not possiblehttps://twitter.com/tomgabion/status/1289857027381002241 …
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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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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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Cantor thought a lot about why ∞ doesn't actually make sense and in order to work with the idea split it into pieces, defining a difference between cardinal and ordinal numbers It's a whole thing that, honestly, the more I think about it the less confident I am at explaining it
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**kicks in the door** SET THEORY BABY!
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Also some of what Cantor was into was on the verge of, if not actually, self-contradictory. Forget the nice order of set theory of the last century, and well, a set of everything is actually self-contradictory. And if there's one thing math flips out over, self-contradiction.
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Also there's a really nice documentary from the BBC back about 18 years ago called Dangerous Knowledge that's about George Cantor, Ludwig Boltzmann, Kurt Godel and Alan Turing and how not only were their ideas radical, but they caused each one to suffer in some way or sense.
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