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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Replying to @lawnerdbarak @arthur_affect and
It isn't part of the Real number set, it is an real as numbers from the Real number set. Real numbers being the combined set of rational and irrational number sets. (I think I am re-calling that correctly).
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Replying to @JohnMarTweets @arthur_affect and
Real numbers are the set of the entire number line, specifically including both rational and irrational numbers, but also including the transcendental irrationals. Like, Q-bar includes infinitely many irrational numbers, but is still the same cardinality as Z
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Replying to @lawnerdbarak @arthur_affect and
But as I was saying, ∞ as used in limits or in a description of the cardinality of a set is instructions, not a closed form quantity. So once you start treating it as a quantity—like cantor did and like set theory does—shit gets weird. Because it isn’t real.
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Right, the whole point is that x+1=x contradicts the basic definition of numbers when we invented them X+1 not being X is how you get the idea of numbers at all
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Replying to @arthur_affect @lawnerdbarak and
So Cantor was being deliberately weird, which is why he had to split up the basic definition of a number into two things in the first place (cardinal and ordinal) and then get weird with it
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Replying to @arthur_affect @lawnerdbarak and
Can you count upwards from infinity to infinity + 1? Depends what kind of infinity (you cannot increase an infinite cardinality in size by counting, but you can choose an "infinitieth" element of the set and then count one up past that)
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