# How do you prove that if a set is countably infinite, then there exists a bijection between from the natural numbers to the set?

A homework question asks the following:

"Let $$X$$ be a countable, not finite set. Show that there is a bijection $$\phi :\mathbb{N}\to X$$."

I was under the impression that this is the definition of what it means for a set to be countable and not finite (i.e. "countably infinite"). Then is this a trivial proof, or is it not the true definition and instead the result of a more subtle axiom?

There is a similar question here, but it involves constructing a bijection from $$X$$ to $$\mathbb{N}$$ rather than from $$\mathbb{N}$$ to $$X$$.

• What is the definition of countable? And what is the definition of finite? Sep 6, 2019 at 3:16
• You're absolutely right OP. Unless you have a different definition to work with, that is the usual definition of countably infinite. Sep 6, 2019 at 3:21
• Context matters. Presumably you are expected to know the definition of countable which lies behind that particular homework problem. My guess will be that this is not exactly the definition, in that context. Sep 6, 2019 at 3:23
• When in doubt, look it up ... and I mean in your textbook or course notes, not on some random site on the Internet. Not even here. Sep 6, 2019 at 3:26
• At the risk of this comment section becoming too long, I should point out that a set $A$ is sometimes defined to be countable if there exists an enumeration (possibly finite) $a_1, a_2, a_3, \ldots$ of the elements of $A$. The exercise would then be to show that the existence of an infinite enumeration is nothing but the existence of such a bijection. Sep 6, 2019 at 3:45