Am I right to think the following?
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?
Thanks, thanks and thanks.