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I had learned that the set is countable if and only if it is finite or countably infinite. We know well that the set $\mathbb{N}=\{1,2,3,4,\dots\}$ is an infinite set. In order to find out if the finite set $X$ is countable, we have to check if there exists an injection $f:X\to \mathbb{N}$ such that $|X|\leq |\mathbb{N}|$. If there is an bijection $f:X\to\mathbb{N}$, then $|X|=|\mathbb{N}|$ and the set $X$ is countably infinite.

Question: Is $\mathbb{N}$ defined to be countable, or has it ever been proven to be countable?

Edit: I know that there's a bijection from $\mathbb{N}\to\mathbb{N}$, and then $|\mathbb{N}|=|\mathbb{N}|$. But this doesn't really answer my question. What if it were $\mathbb{R}\to\mathbb{R}$ instead? I just want to know where the idea come from to decide that $\mathbb{N}$ is countable. And then we talk about other sets, like $(0,1)$ and $\mathbb{R}$. Have the mathematicians chosen that the $\mathbb{N}$ is defined to be countable? I can not find any sources that explain about it. Many definitions about the countable set is including the "countable" set $\mathbb{N}$, for example, a set $X$ is countable if and only if $X\leq \mathbb{N}$.

Edit: I was really bad at expressing what I meant. I'll write down short how I found the "answer". In reality, I misunderstood definition completely that made me question something rubbish. The definition is

A set $X$ is countable, if and only if $|X|\leq \mathbb{N}$.

This doesn't say anything about $\mathbb{N}$ (which is what the question was about), but only about $X$. If letting $X=\mathbb{N}$ and that there's an injection from $X$ to $\mathbb{N}$ would imply that the set $X$ is countable. That's it.

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    $\begingroup$ Can you think of a bijection from $\mathbb{N}\rightarrow\mathbb{N}$? $\endgroup$ – Ben Sheller Aug 16 '15 at 23:41
  • $\begingroup$ @BenS. Oh, never thought about it until now. Thanks! $\endgroup$ – UnknownW Aug 16 '15 at 23:44
  • $\begingroup$ @BenS. If there is a bijection from $\mathbb{N}\to \mathbb{N}$, then $|\mathbb{N}|=|\mathbb{N}|$, but how is it decided that the set $\mathbb{N}$ is countable? What if there is a bijection from $\mathbb{R}\to\mathbb{R}$? I know well that $\mathbb{R}$ is not countable if I accepted that $\mathbb{N}$ is countable. I just want to know why $\mathbb{N}$ is countable to begin with, or where this idea come from. $\endgroup$ – UnknownW Aug 17 '15 at 1:23
  • $\begingroup$ I think that usually people define something to be countable if it has the same cardinality as a subset of $\mathbb{N}$ (which is what you have written above). One of the motivations behind calling $\mathbb{N}$ or a subset of $\mathbb{N}$ is that you can hope to label the elements (if you have an infinite amount of time): 1,2,3,4,.... For something like $\mathbb{R}$, you can't ever hope to do that: For any number $x\in\mathbb{R}$ which we assign the label $1$, we have such a huge amount of numbers between $x$ and any other numbers, that there is no hope of doing any counting! $\endgroup$ – Ben Sheller Aug 17 '15 at 2:32
  • $\begingroup$ One of the most surprising things is that $\mathbb{Q}$ is countable but $\mathbb{R}$ is not...you can read about that sort of thing here for example: link $\endgroup$ – Ben Sheller Aug 17 '15 at 2:33
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Countably infinite sets are "the same size as" $\Bbb{N}$, which means that there's a bijection (a one-to-one correspondence) between them and the natural numbers. The identity function $\Bbb{N} \to \Bbb{N}$ is, of course, a bijection, showing that the set of natural numbers is countably infinite.

Cardinality ("size") of a set is a type of equivalence relation on sets: two sets are equivalent if they have the same cardinality. The reflexive property is what your question is about. It's worth trying to prove the other two properties: symmetry and transitivity.

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Countable infinite is just the "firsth class of infinity", and the collection of natural numbers is the prototype example. Need for uncountable sets was the proper theory of real numbers, i.e. the point set theoretical understanding of a geometrical line. Cardinality is just a method to classify different sizes of sets since due to Cantor's diagonal argument there are different types of infinities.

After that mathematicians started to do what they usually do: they go axiomatics.

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