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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The *rules of arithmetic* say this -- numbers are *defined* by the fact that if you add 1 or subtract 1 from a number, you get a different number, if "∞" breaks this rule, it's not a number at all and you can't use it
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So people's intuitions about this are correct but they don't take it far enough Caught between the dueling intuitions between "Well the answer to this question can't be any ordinary number" and "It must HAVE an answer though"
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All new forms of math are built on that second part "Okay, I get that this is a stupid-ass question and Pythagoras or whoever would've just said 'shut the fuck up' if I asked him but WHAT IF you COULD add up numbers an infinite number of times"
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All summations of infinite series are based on making up new rules and saying "Okay let's pretend you can do this, what happens if you do, what new stuff do you discover if we just fuck around and act like this makes sense"
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So like, let's be clear This classic series: 1+1/2+1/4+1/8+1/16... Is, from a pure old-school POV, just as bad as the other one Even though this one looks like it has an answer (it adds up to 2 in the end)
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Replying to @arthur_affect
Gah! I hate this math. How do you ever get to 2? Because there's no fraction so small that half of it is zero. Are we saying there's some denominator so large that if it were x then 1/x = 0? Because that's the only way I can see this = 2. I am not suited to math like this.
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Replying to @TheCheekyGinger
Well, for many years we were trying to avoid saying that, and just define the idea of a "limit" without saying actual "infinite" and "infinitesimal" numbers exist But now there's non-standard analysis which says that those are okay
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Replying to @arthur_affect @TheCheekyGinger
But yeah ignoring that noise, the idea of the delta-epsilon definition of a limit is you don't ever need to "finish" the series (you can't, it takes infinitely long)
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Replying to @arthur_affect @TheCheekyGinger
You just need to ask the question "Every time I go one step further, is there a specific number I am getting closer to" And in this case yes, you can see that it's 2, so the answer to the question is 2
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But yes, as I said in the main thread, this idea when first brought up was very controversial
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