Talk:Real Numbers are Uncountably Infinite/Set-Theoretical Approach: Proof 1

"The above needs to be established as a result. Our definition is based on the fact that there is an injection the other way."


 * If there exists an injection the other way, we can send any element not in the image to, say, $1$. However, the converse asserts that any surjection into $\N$ permits a section (i.e., an injection the other way). This statement probably needs some choice axiom (not necessarily the full-fledged AC, maybe ACC will do). --Lord_Farin (talk) 08:20, 5 October 2012 (UTC)


 * I think it can be proved using the Zermelo–Fraenkel axioms alone, because $\N$ is well-ordered. Hence there exists a choice function on $\powerset \N \setminus \set \O$, so we can use the proof in Surjection iff Right Inverse/Proof 1 without invoking any choice axioms. --abcxyz (talk) 16:00, 5 October 2012 (UTC)


 * Sounds reasonable; would you care enough to try and locate it, and if not present, put it up and prove it? I'm not confident I could do it without flaws. --Lord_Farin (talk) 16:50, 5 October 2012 (UTC)


 * Okay, I'll have to see when I could do that. How about "Surjection from Natural Numbers iff Right Inverse"? --abcxyz (talk) 17:34, 5 October 2012 (UTC)


 * Sounds good. --Lord_Farin (talk) 07:57, 6 October 2012 (UTC)


 * I've modified the Definition:Countable definition to include the equivalent definition of the existence of a surjection. --Nitraat (talk) 00:12, 12 November 2012 (UTC)


 * I've already included a link to the page proving the equivalence (see Also See section at the bottom). --abcxyz (talk) 00:17, 12 November 2012 (UTC)