Talk:Power Set of Natural Numbers is not Countable

It occurs to me that the lack of a bijection isn't sufficient here, since our definition of countable is that there is an injection to $\N$. It's entirely possible to have a set inject into $\N$ without there being a a bijection between the two. For example, $\{1\}$ clearly injects into $\N$ by the identity function and is thus countable, but there is clearly no bijection from $\N$ to $\{1\}$. Thoughts? --Alec (talk) 10:59, 10 March 2011 (CST)


 * From the defs:


 * An infinite set $X$ is countable $\iff$ there exists a bijection $X \leftrightarrow \N \qquad (1)$


 * Therefore is there is no bijection with $\N$, an infinite set is uncountable.


 * Perhaps $(1)$ should be proved, but `surjects onto $\N$' is pretty much the definition of infinite. --Linus44 12:24, 10 March 2011 (CST)


 * Feel free to rewrite the definitions and proofs of infinitude and countability so they're consistent and use minimal assumptions. I tried that a while back and got emmerdé. --prime mover 15:27, 10 March 2011 (CST)

They look ok to me, it's just a matter of switching between formulations; I think it'll crop up somewhere however they're formulated. --Linus44 16:04, 10 March 2011 (CST)