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.
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.
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.
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.
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.