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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It comes from the idea of treating the infinitesimal as actually a thing, like there really is such a thing as "infinitesimally more than 4" (4+ε) and therefore you can take the reciprocal of "infinitesimally more than 4" and get "1/4+infinity"
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But just as the ε isn't actually 0, because 4+0 would just be 4, the "infinity" isn't actually the old school (contradictory) idea of this objective highest number that exists ∞ that is the result of 1/0
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I once volunteered to explain Hilbert's Hotel for a job interview where the prompt was to teach the interviewers something.
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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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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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