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 @lawnerdbarak and
The problem seems to me rather that one starts with vage preconceptions of ∞ as some definite quantity, an idea that, I think, has never worked.
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Replying to @hllizi @lawnerdbarak and
Well that's just it, Cantor did invent rules for treating ∞ as a "definite quantity" and they do work, they just don't work the way you'd expect working with real numbers (because it's not a real number)
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Replying to @arthur_affect @hllizi and
Aleph-null, the first infinite cardinal number (the size of the set of all natural numbers), actually does do all the paradoxical stuff people think of "infinity" doing It stays the same size if you add 1 to it, if you multiply it by 2, if you square it, etc
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Hence you can't, in fact, usefully use it as "the biggest number" in normal math problems, you immediately run into contradictions
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Replying to @arthur_affect @hllizi and
As I learned from Reg Cathy when I was like 5 It’s not a value, it’s an instructionhttps://youtu.be/mPUyJjm6QnI
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