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I'd like to know if this proof of $\mathbb{R}$ being uncountable is correct. (It came to my mind while reading this proof.)

$\mathbb{R}$ is countable iff there is a surjective function $f:\mathbb{N} \mapsto \mathbb{R}$. In order for such a function to exist, it must be true that $\mathbb{N} \supseteq \mathbb{R}$. But that is false, which means $\mathbb{R}$ is not countable.

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    If this were true, then certainly the uncountability of $\mathbb{R}$ would have been proven much earlier in history.2011-10-16

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No, that does not work. There is no definition that says that the image of a surjective function must be a subset of the domain. For example, $f(x)=\pi$ is a perfectly good surjective function $\mathbb N\to\{\pi\}$, but $\mathbb N\not\supseteq\{\pi\}$.

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No, that argument doesn't work. For example, take the set \mathbf{N}=\{1',2',3',\ldots\}. There is a surjective function $f:\mathbb{N}\to \mathbf{N}$, defined by f(n)=n' for all $n\in\mathbb{N}$, which demonstrates that $\mathbf{N}$ is countable. However, as the symbols n' are not natural numbers, $\mathbf{N}\not\subseteq\mathbb{N}$.

In general, if you are wondering whether such a statement is correct, ask yourself if what you mean would still be correct if you took the elements of your set and put ' on all of them, or painted them all red, or replaced each element with its identical twin (each of my professors has had their own funny way of saying this).