So, @deathbob and myself were discussing whether the real numbers have the same cardinality as any interval of the real numbers. That is, if you pick an interval (say [0,1]), is it true that the cardinality of that interval is the same as that of the entire set of real numbers? Here's a defense of that idea.
Theorem 1: Pick any two intervals of the real numbers R. Then their cardinality is the same.
Proof: Let an interval
s < t) be the set of all real numbers between
t, inclusive. To prove that the cardinality is equal, we need to show that you can write a one-to-one correspondence between any two such intervals -- say,
There are lots of ways to do this, but a simple way to do it is just to map them linearly. Let's say you have the two intervals [0,1] and [0,2]. Take any real number in [0,1] -- say, 0.781. Now double it. You have just mapped it to a real number in [0,2], and you can