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] (with s < t) be the set of all real numbers between s and 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, [s,t] and [u,v].
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 do this for every number in [0,1]. Therefore these two intervals must have precisely the same cardinality.
Consider another example. If you have the intervals [2,4] and [16,32], then some examples for our function f would return f(2) = 16, f(3) = 24, and f(4) = 32. Likewise it would also return f(2.1) = 16.8 and f(3.5) = 28.
More generally, if you have two finite intervals [s,t] and [u,v], you can map the reals in a one-to-one way between them with
v - u
f(x) = ----- * (x - s) + u
t - s
Using our previous example, you can see how f(2.1) works with [2,4] and [16,32]:
32 - 16
f(2.1) = ------- * (2.1 - 2) + 16
4 - 2
= 16/2 * 0.1 + 16
= 8 * 0.1 + 16
= 0.8 + 16
= 16.8
Since this maps the reals in a one-to-one way, we've shown that any two intervals of reals have the same cardinality.
Theorem 2: The cardinality of any interval is equal to the cardinality of the reals.
Proof: This time we need to construct a one-to-one correspondence between the reals and any specific interval in it. For our interval, let's pick (-1, 1). Just as before, there are lots of functions that will work for this, but let's pick something simple -- say:
x
f(x) = -------
|x| + 1
This maps any real number x and produces another real number which is between -1 and 1. For instance, f(5000) = 5000/5001; f(-296) = -296/297.
Therefore, because this can pair the entire set of real numbers with any number in the interval (-1, 1), the cardinality of the real numbers must be the same as the cardinality of the interval (-1, 1). And earlier we showed that the cardinality of any two intervals is the same, so the cardinality of the real numbers is the same for any interval in it. QED.