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orthonormal 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.
I think that part of the difficulty (and part of the reason that certain people call themselves infinite set atheists) stems from the fact that we have two very basic intuitions about the quantity of finite sets, and it is impossible to define quantity for infinite sets in a way that maintains both intuitions.
Namely, you can have a notion of quantity for which
(A) sets that can be set in some 1-to-1 correspondence will have the same quantity,
OR a notion of quantity for which
(B) a set that strictly contains another set will have a strictly larger quantity.
As it turns out, given the importance of functions and correspondences in basic mathematical questions, the formulation (cardinality) that preserves (A) is very natural for doing math that extends and coheres with other finite intuitions, while only a few logicians seem to toy around with (B).
So it may help to realize that for mainstream mathematics and its applications, there is no way to rescue (B); you'll just need to get used to the idea that an infinite set and a proper subset can have the same cardinality, and the notion that what matters is the equivalence relation of there existing some 1-to-1 correspondence between sets.
(B) is roughly measure theory, innit?
Yes, for some value of "roughly".
(A value of "roughly" that encompasses sets of measure zero is what I had in mind.)
My problem doesn't arise only when comparing sets such that one strictly contains another. I can "prove" to myself that there are more rational numbers between any two integers than there are natural numbers, because I can account for every last natural number with a rational between the two integers and have some rationals left over. I can also read other people "proving" that the rationals (between two integers or altogether, it hardly matters) are "countably infinite" and therefore not more numerous than the integers, because they can be lined up. I get that the second way of arranging them exists. It's just not at all clear why it's a better way of arranging things, or why the answer it generates about the relative sizes of the sets in question is a better answer.
If you come up with a different self-consistent definition of how to compare sizes of sets ("e.g. alicorn-bigger"), that would be fine. Both definitions can live happily together in the happy world of mathematics. Note that "self-consistent definition" is harder than it sounds.
There are cases where mainstream mathematical tradition was faced with competing definitions. Currently, the gamma function is the usual extension of the factorial function to the reals, but at one time, there were alternative definitions competing to be standardized.
http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html
Another example: The calculus was motivated by thought experiments involving infinitesimals, but some "paradoxes" were discovered, and infinitistic reasoning was thought to be the culprit. By replacing all of the arguments with epsilon-delta analogs, the main stream of mathematics was able to derive the same results while avoiding infinitistic reasoning. Eventually, Abraham Robinson developed non-standard analysis, showing an alternative, and arguably more intuitive, way to avoid the paradoxes.
http://en.wikipedia.org/wiki/Non-standard_analysis
Thanks for that super-interesting link about factorial-interpolating functions!
The trouble is that with a little cleverness, you can account for all of the rationals by using some of the natural numbers (once each) and still have infinitely many natural numbers left over. (Left as an exercise to the reader.) That's why your intuitive notion isn't going to be self-consistent.