LRT: The idea of adding to infinity ("you can't do it, you'd still be adding when the sun went out") has me thinking of that short story I wrote where an immortal being trapped in an endless void tries to count to infinity to pass the time and eventually runs out of named numbers
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Let's just assume you could "pick a random real number" (axiom of choice) A truly random one, with nothing special about it like pi How would you name it, how would you know what it was
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Its decimal representation is an infinite series of digits You'll never get to the end of it, you can never stop writing it, and because it's got nothing special about it, nothing tells you what the next one will be
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