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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Replying to @arthur_affect @hllizi and
Brb, gonna write in a character named “Aleph-null” to make myself seem clever.
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Replying to @beetlefella101 @hllizi and
That's an old meme, there's many many such characters
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Replying to @arthur_affect @beetlefella101 and
Aleph null bottles of beer on the wall Aleph null bottles of beer Take one down Pass it around Aleph null bottles of beer on the wall
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Aleph-null bottles of beer on the wall Aleph-null bottles of beer Take aleph-null down Pass them around Aleph-null bottles of beer on the wall
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Replying to @arthur_affect @lawnerdbarak and
This verse is logically equivalent to the previous one! Which is very frustrating
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Replying to @arthur_affect @beetlefella101 and
It works better if it’s “take them all down, pass them around”
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