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I think I understand Cantor's diagonalization argument, but I'm trying to wrap my head around this consequence of it.

Let's suppose I pick an arbitrary interval, say, $[5, 6]$. Is it true that the number of reals in this interval is the same as the number of reals in $\mathbb R$? How can that be when $\mathbb R$ is clearly bigger than that interval?

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    It just means there is some one-to-one correspondence between the interval and all of $\mathbb{R}$. Yes, it is quite bizarre, but you've actually encountered something like it before. The function $\frac{1}{x}$ on $[-1,1] \setminus \{0\}$ shows a correspondence between the domain and the vast majority of $\mathbb{R}$.2011-11-03
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    @AustinMohr Thank you for the clarification, I edited accordingly.2011-11-03
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    "the same number of elements" is loose speaking, infinite sets do not really have a "number of elements". If by that you mean the cardinality, yes, that's true. And that (an infinite set can have the same cardinality of a proper subset of itself) is a basic "paradoxical" property of infinite sets, rather more basic than then diagonalization argument (it applies also to countable infinite sets). See the Hotel paradox http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel2011-11-03
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    @Austin, you mean $(-\frac\pi2,\frac\pi2)$, right?2011-11-03
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    You have to reconcile yourself with a different notion of "bigger." For example, under Cantor's definition of "same size," any infinite set of natural numbers is the same size as the natural numbers. So there are as many perfect squares, for example.2011-11-03
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    Or, a better example, the function $\tan(x)$ is a one-to-one correspondence between $(−\frac{\pi}{2},\frac{\pi}{2})$ and all of $\mathbb{R}$.2011-11-03
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    @leonbloy: While it does not coincide with the layman idea of "number of elements", to say that two sets have the same cardinality is to say that both have the same number of elements. It is perfectly fine to say "I have in $A$ exactly $\aleph_{42}$ many elements."2011-11-03

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