Take the set of all real numbers from zero to one inclusive. Call it One. Take the set of all real numbers from zero to two inclusive. Call it Two. Let F be the function from One to Two such that for all x in One, F(x) = 2x. The claim is: F is a one-to-one correspondence from One to Two.
My reasoning is that, since every member of Two determines a unique member of One through division by two, and since no two members of Two determine the same member of One through division by Two, and since division by two is the inverse of multiplication by two, it follows (I think) that F’ is one-to-one. Since F is one-to-one and F’ is one-to-one, it follows (if I recall/understand correctly) that F is a one-to-one correspondence.
A related question. Even if I’m not right that multiplication by two constitutes a 1-1 correspondence between the two, still, we know there is some such correspondence. Is this fact part of why it is supposed to be possible to cut a line into pieces in such a way as to be able to reassemble them to form a line twice as long (or indeed of any arbitrary length)? For if there is a 1-1 correspondence between the points on a line of length 1 and a line of length 2, then it should be logically possible to “disassemble” the line of length 1, shift the points around in a way such that each ends up in its corresponding place in the line of length 2, and so “reassemble” the line in a way that it ends up having twice its length, right?
Finally, does the theorem about being able to similiarly double the volume of a sphere follow directly and trivially from this, or is there some complication involved there?
Upon further research, I think you’re thinking of the Banach–Tarski paradox. But I don’t think it applies to one-dimensional intervals, so your paragraph about disassembling the interval [0,1] and rearranging it into the interval [0,2] is (as far as I know) incorrect, assuming you’re not stretching the pieces, which is what F(x) = 2x does.
Yep, looks like Banach-Tarski is much more complicated. It says something different than I was thinking, anyway. It says the sphere can be cut into a finite number of pieces rearrangeable into two spheres. I haven’t said anything about the corresponding possibility of rearranging a line cut into a finite number of pieces.
I think I’m right, though, that you could cut the line into an infinite number of pieces and rearrange them into a line twice as long. But I think that if I’m right about that, it’s uninteresting once you understand it.
I mean this as a serious question: What is stretching?
-FrL-
BTW I knew I was remembering something about how you’re supposed to be able to rearrange a line into a line of twice the length. Turns out I had in mind the (second listed) Hausdorff Paradox. And there is a relation between it and Banach-Tarski, and that relation is this apparently in its entirety: There is a link from the Banash-Tarski wiki entry to the Hausdorff wiki entry. :smack:
I don’t know whether there’s an official definition, but by “not stretching the pieces” I meant a transformation that preserves the measure of the pieces (= their lengths, if they are intervals).
If I understand the Wiki article you linked to correctly, the pieces that Hausdorff chops the unit interval into are not measurable, so you can’t really say that his rearrangement preserves measure.
(I am not an expert on any of this. I know we have some good mathematicians here on the SDMB; hopefully a set of them of measure > 0 will stop in and contribute.)
What you outlined in your OP is the proof that there are the same number of points in a short line segment as in a long one. But this isn’t particularly relevant to the Banach-Tarski paradox, since that number of points is infinite. In fact, almost any interesting geometric object (anything which includes pieces of at least one dimension, more or less) has that same number of points.
This is one example of the way that manipulating infinities does not correspond to the common sense notions we have.
As **Chronos ** says, a line an inch long, a one-inch cube, and the entire universe contain exactly the same infinite number of points. However, rather than implying you can rearrange one into the other, it’s better to think of it as not being able to cut the one-inch line into an infinite number of pieces in the first place. That’s the part that has no meaning and doesn’t have a real world correspondence.
