There's a real between any two rationals, a rational between any two reals, but more reals than rationals? The following statements are all true:


*

*Between any two rational numbers, there is a real number (for example, their average).

*Between any two real numbers, there is a rational number (see this proof of that fact, for example).

*There are strictly more real numbers than rational numbers.


While I accept each of these as true, the third statement seems troubling in light of the first two.  It seems like there should be some way to find a bijection between reals and rationals given the first two properties.
I understand that in-between each pair of rationals there are infinitely many reals (in fact, I think there's $2^{\aleph_0}$ of them), but given that this is true it seems like there should also be in turn a large number of rationals between all of those reals.
Is there a good conceptual or mathematical justification for why the third statement is tue given that the first two are as well?
Thanks!  This has been bothering me for quite some time.

Here's an attempt at a moral justification of this fact. One (informal) way of understanding the difference between a rational number and a real number is that a rational number somehow encodes a finite amount of information, whereas an arbitrary real number may encode a (countably) infinite amount of information. The fact that the algebraic numbers (roots of polynomial equations with integer coefficients) are countable suggests that this perspective is not unreasonable. 
Naturally, when your objects are free to encode an infinite amount of information, you can expect more variety, and that is ultimately what causes the cardinality of $\mathbb{R}$ to exceed that of $\mathbb{N}$, as in Cantor's Diagonal Argument. However, because real numbers encode a countable amount of information, any two distinct real numbers disagree after some finite point, and that is why we may introduce a rational in the middle. 
All in all, this is seen to boil down to the way we constructed $\mathbb{R}$: as the set of limit points of rational cauchy sequences. This is because a limiting process is built out of "finite" steps, and so we can approximate the immense complexity of an uncountable set with a countable collection of finite objects.
