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This question already has an answer here:

How can the real, integer, and rational number sets be infinite, yet, they aren't all the same size?

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marked as duplicate by user228113, Community Dec 16 '15 at 22:25

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  • $\begingroup$ Look up "cardinality" and "cardinal numbers" using google. Also "countable set" would be good, and maybe "Cantor's diagnoan proof." BTW integers, rationals same size in cardinality. $\endgroup$ – coffeemath Dec 16 '15 at 22:16
  • $\begingroup$ And this, and this... $\endgroup$ – user228113 Dec 16 '15 at 22:23
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Because of Cantor's theorem (the diagonal construction).

Given the definitions of what it means for two infinite sets to be of the same size, and of what it means for one set to be at least as large as another, $\Bbb N$ and $\Bbb Q$ are of the same size, because we can prove that there is a bijection between them.

However, Cantor's theorem shows that no 1-1 function $\Bbb N\to \Bbb R$ can exhaust all of $\Bbb R$ — it can't be onto $\Bbb R$, so $\Bbb N$ is strictly smaller than $\Bbb R$.

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