What I got from this post is that the weirdness here comes in, not so much from the Hilbert's Hotel phenomenon about cardinalities and sets that can be put into correspondence with their own proper subsets, but from looking more deeply at something relatively familiar and something that we ordinarily use to tame infinities: volume.
Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points.
This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments).
While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?
>...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1...
>...even though it contains infinitely many points.
There is the division by infinity.
Cut the volume in half. Still has infinite points. infinity/2 = infinity.
1 = 1/2, or 1=2 if you can cancel those.
Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to infinitely many of them to have a volume other than zero. And you're back to infinity points * (1/infinity) vol of a point.
So yeah, no way I can see to have any set of points have a well defined volume because the volume of a point is a division by infinity, unless the set is finite in which case it is zero (or whatever you define a finite number divided infinitely to be - define it to be gerald if that helps? - Mathematical immaturity on display right there).
Yes, I see your point (no pun intended), but I would still maintain that this issue didn't necessarily bother mathematicians in the days before formalized set theory. The issue you mention is a reason that any definition of volume should not be based only on set cardinality, since there are those one-to-one correspondences with proper subsets (for infinite sets of points), yet volume should not be preserved by cutting something in half, or doubling something, in an ordinary way.
That is, if X is the unit line segment and Y is a pair of unit line segments joined end-to-end, we want the measure function µ to obey µ(Y) = 2µ(X), even though |X|=|Y| in set theory. And even though that's weird, formalized mathematics didn't get existentially ambitious enough to make anything truly bizarre out of it, I suppose, from Euclid all the way up until Vitali!
Yeah for me it simply doesn't. It isn't false, it just doesn't even make sense.
Define X and Y as you did but make one of the segments joined to make Y /be/ X.
Now if it is ever meaningful to have a one to one correspondence when infinity is lurking about surely the points on X correspond exactly, one to one, each with itself. Once all the points of X are accounted for, clearly half of Y remains, there is no other way. There is simply no point you can select in X that is free to correspond to the second half of Y if you select corresponding to itself in preference. You can never, ever select a point corresponding to the second half of Y if you admit preference to corresponding to itself in selection.
But X has an inexhaustible number of points to chose from by definition and so does Y. So if you start picking points at random from Y and matching them up to a previously unused point from X you can do that forever and never exhaust either of the sets of points. Thus X and Y correspond one to one and no points remain of either X or Y. And there are the same number of integers as there are integers that are even.
1=2 QED The walls are different measured heights but that doesn't matter because we've proved it. Yeah ok, but maybe it does to the poor sap who needs a useful roof?
In that case, I would say you're a finitist (which, as the name seems to suggest, is fine).
The set theory approach gives seemingly clear answers to all of these questions by talking about domains and ranges of functions, rather than talking about what is or isn't an "inexhaustible number". There are potentially different correspondences available, represented by different functions; some possible correspondences may follow the "if you select corresponding to itself in preference" pattern, while others don't. = and ≤ for cardinalities are defined using existence of certain functions between sets.
However, you don't have to believe that any infinite sets exist or that we should be allowed to quantify over them, or that we should attempt to define cardinality for infinite sets at all. Still, as Dana Scott said in a related context, "if you want more, you have to assume more".
Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points.
This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments).
While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?