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saturn comments on The Lifespan Dilemma - Less Wrong
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You are not the first person to try to explain this to me, but it doesn't seem "surprising", it seems like everybody is cooperating at pulling my leg. Since I'm aware that such a conspiracy would be impractical and that I am genuinely terrible at math, I don't think that's actually happening, but the fact remains that I just do not get this (and, at this point, no longer seriously entertain the hope of learning to do so). It is only slightly less obvious to me that there are more numbers between 0 and 2 than 0 and 1, than it is that one and one are two.
To put it a little differently, while I can understand the proofs that show how you may line up all the rationals in a sensible order and thereby assign an integer to each, it's not obvious to me that that is the way you should count them, given that I can easily think of other ways to count them where the integers will be used up first. Nothing seems to recommend the one strategy over the other except the consensus of people who don't seem to share my intuitions anyway.
Imagine A is the set of all positive integers and B is the set of all positive even integers. You would say B is smaller than A. Now multiply every number in A by two. Did you just make A become smaller without removing any elements from it?
...Okay, that's weird! Clearly that shouldn't work. Thanks for the counterexample.
It gets even worse than that if you want to keep your intuitions (which are actually partially formalized as the concept natural density). Imagine that T is the set of all Unicode text strings. Most of these strings, like "๐พโจ๊ โงฬโฉถ๐", are gibberish, while some are valid sentences in various languages (such as "The five boxing wizards jump quickly.", "print 'Hello, world!'", "แผฯฯฮฑฯฮฟฯ แผฯฮธฯแฝธฯ ฮบฮฑฯฮฑฯฮณฮตแฟฯฮฑฮน แฝ ฮธแฝฑฮฝฮฑฯฮฟฯฮ", or "ืืงืจืืชื ืืฉื ืืืืืื ืืื ื ืืงืจื ืืฉื ืืืื ืืืื ืืืืืื ืืฉืจ ืืขื ื ืืืฉ ืืื ืืืืืื ืืืขื ืื ืืขื ืืืืืจื ืืื ืืืืจ"). The interesting strings for this problem are things like "42", "22/7", "e", "10โโ(10โโ10)", or even "The square root of 17". These are the strings that unambiguously describe some number (under certain conventions). As we haven't put a length limit on the elements of T, we can easily show that every natural number, every rational number, and an infinite number of irrational numbers are each described by elements of T. As some elements of T don't unambiguously describe some number, our intuitions tell us that there are more text files than there are rational numbers.
However, a computer (with arbitrarily high disk space) would represent these strings encoded as sequences of bytes. If we use a BOM in our encoding, or if we use the Modified UTF-8 used in Java's DataInput interface, then every sequence of bytes encoding a string in T corresponds to a different natural number. However, given any common encoding, not every byte sequence corresponds to a string, and therefore not every natural number corresponds to a string. As encoding strings like this is the most natural way to map strings to natural numbers, there must intuitively be more natural numbers than strings.
We have thus shown that there are more strings than rational numbers, and more natural numbers than strings. Thus, any consistent definition of "bigger" that works like this can't be transitive, which would rule out many potential applications of such a concept.
EDIT: Fixed an error arising from my original thoughts differing from the way I wanted to explain them