Talk:Domain of Injection to Countable Set is Countable

What exactly is proven here? First, it is assumed that $X$ is infinite. If there exists an injection $f : X \to \N$ then $X$ is countable, by definition. So the assumptions are equivalent to the condition that $X$ is countably infinite. Is the goal to prove that $X$ is equivalent to $\N$? If so, maybe this page should be called "Countable Infinite Sets are Equivalent"? But that's not even proven on this page. --abcxyz 00:02, 5 April 2012 (EDT)

It proves that if there exists an injection from $X$ to $\N$ then $X$ must be countable. --prime mover 01:19, 5 April 2012 (EDT)

Well, isn't that just the definition of countability? --abcxyz 01:22, 5 April 2012 (EDT)

Good call. I have a feeling I got this from a mathematical logic course which might have glossed over the concepts of countability so as to get on with doing other things instead. As such there's plenty of other pages that link to this. Can we leave it in for the moment (as it's not technically incorrect) until I have found time to go back through my source works to see where their definition of countability actually came from in the first place? --prime mover 01:33, 5 April 2012 (EDT)

All right, let's do other things for now. --abcxyz 01:35, 5 April 2012 (EDT)