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
Well - a number that you can do math with. I'm not sure that qualifies as a 'definite quantity' though.
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Replying to @mssilverstein @arthur_affect and
I'd certainly regard alelph 0 as a definite quantity as I meant it. My point is just that, as you climb up the power set hierarchy, you get greater quantities but - of course - all of them are infinite. So there is not just one single quantity "infinite".
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Replying to @hllizi @mssilverstein and
To put it like this: "larger than three" is not a definite quantity. And in a similar sense, the exploration of the transfinite revealed that - if you want to accept such a thing at all - neither is "greater than every natural number".
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Yeah fair enough, although what Cantor was trying to do was specifically establish that there was a specific number that referred to all of a specific "kind of infinity"
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Replying to @arthur_affect @hllizi and
Like that aleph-null is the size of the set of all natural numbers, and all even numbers, and all prime numbers, and all rational numbers, and all the powers of ten You can use the = sign to say these are all the same number
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Replying to @arthur_affect @mssilverstein and
OK, I think now we're on the same page again.
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