# 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.

• "It seems like there should be some way to find a bijection between reals and rationals given the first two properties." I highly encourage you to actually attempt to do this and see what goes wrong. – Qiaochu Yuan Mar 26 '12 at 1:23
• The answer, in a nutshell: real numbers correspond to sets of rationals rather to rationals, and there are a lot more sets than rationals. – Asaf Karagila Mar 26 '12 at 1:36
• Also, very related question here: math.stackexchange.com/questions/18969 I'm not sure if it is a duplicate or not, but it's quite close. – Asaf Karagila Mar 26 '12 at 1:38
• @Asaf: (This addresses your nutshell comment.) Maybe, but not all these sets at all are required, so I am not sure I want to buy the argument as is. – Did Mar 26 '12 at 7:58
• @Didier: I have a bijection to guarantee that all sets are playing. Also, this argument is not for sale! – Asaf Karagila Mar 26 '12 at 8:09

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.