# About the cardinality of natural numbers [Solved]

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.

• Can you think of a bijection from $\mathbb{N}\rightarrow\mathbb{N}$? – Ben Sheller Aug 16 '15 at 23:41
• @BenS. Oh, never thought about it until now. Thanks! – UnknownW Aug 16 '15 at 23:44
• @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. – UnknownW Aug 17 '15 at 1:23
• 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! – Ben Sheller Aug 17 '15 at 2:32
• 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 – Ben Sheller Aug 17 '15 at 2:33

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.