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Alicorn comments on The Lifespan Dilemma - Less Wrong
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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.