You can't say x="infinity" or ∞ because in ordinary arithmetic that's not a number, it's a meaningless word
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Replying to @arthur_affect
I’d argue against meaningless. It’s very meaningful, but it’s not an actual attainable thing. If infinity is meaningless, imaginary numbers (sqrt(-1)) are meaningless since it cannot be found in nature.
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Replying to @KHMakerD
Oh, I'll backtrack quickly on this one -- "infinity" *as most people use it* doesn't have very *much* meaning because they don't define what they're talking about Cantor, who spent his life studying the concept of infinity, jumped very quickly past ∞ to infinities, plural
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Replying to @arthur_affect @KHMakerD
So, in other words, he went to infinity...and beyond?
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Replying to @beetlefella101 @KHMakerD
There's an infinite number of infinities He even went ahead and defined "absolute infinity" (Ω) and said that's the name for an infinity so big that none of the properties he'd just mathematically explored applied to it
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(Sort of supporting the point Spengler would later make about "infinity" being a peculiarly European Christian cultural concept, Cantor was a *very* devout Christian who saw his mathematics as a logical extension of the whole "Can God make a rock so big he can't lift it" stuff)
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He was kind of caught between worlds here, because the other mathematicians -- most of whom weren't particularly devout -- often thought of his work as a weird waste of time, while the Church didn't understand what he was talking about and thought it sounded like witchcraft
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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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