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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.

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"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
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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
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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
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2 Answers

Maybe this helps: if two rationals agree to (say) 73 decimal places, then you can find reals between them by making your reals have the same first 73 decimals, then be careful with the 74th place, but then do anything you want with all the places after that (this isn't exactly right, if the 74th places differ by 1, but that's easy enough to fix).

Now if you have two reals that agree to 73 places, you can find rationals between them by making your rationals have the same first 73 decimals, then be careful with the 74th, but then you have to make sure the rest of it is eventually periodic (or terminates). That restriction on the rationals, which wasn't there for the reals, may help you see why there are more reals (even between two rationals that agree to 73 places) than rationals.

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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.

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