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Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs of irrationals. Therefore the number of irrationals is countable since the cardinality of $2\mathbf{N}$ is $\mathbf{N}$.

I don't know why I came across this logic since I know the irrationals are uncountably infinite, but I don't see the hole in my logic.

The Substitute
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    To make this argument work, you'd need a one-to-one correspondence between rational and irrational numbers. Part of the challenge of finding such a correspondence lies in the fact that between any to irrationals there is not just one rational number, but infinitely many. – Michael Hardy Jul 04 '12 at 22:32
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    It is true that between any two irrational lies a rational, but the same rational can separate many pairs of irrationals! – dtldarek Jul 04 '12 at 22:33
  • It might help if you try to turn this heuristic into a formal proof and see where it goes wrong. If you are trying to construct a map from the irrationals to the rationals, you should have a hard time ensuring that the mapping is one-to-one. – guy Jul 04 '12 at 22:48
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    http://math.stackexchange.com/questions/18969/why-are-the-reals-uncountable – Andrés E. Caicedo Jul 04 '12 at 23:59
  • The basic problem here is surely with your first statement, "Between any two irrationals lies a rational, by the Density of the rationals in the real number system." There is a big difference between "There exists an $x$ such that so-and-so holds of $x$" and "There exists a unique $x$ such that so-and-so holds of $x$". Density allows you to conclude the first, but not the second. – Benedict Eastaugh Jul 08 '12 at 12:09

2 Answers2

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The claim that between two rationals there is only one irrational is false. In fact between two rationals there are many irrationals, so you will have to map a lot of irrationals to the same pair (for most pairs too).

Therefore your proof does not constitute of a bijection, or even a well-defined function. This is a common mishap with infinite objects, though. They tend to get very confusing!

Asaf Karagila
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To make this argument work, you'd need a one-to-one correspondence between rational and irrational numbers. Part of the challenge of finding such a correspondence lies in the fact that between any to irrationals there is not just one rational number, but infinitely many.

Michael Hardy
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