Talk:Rational Numbers are Countably Infinite

Intuitive Proof Wrong
The intuitive proof doesn't seem to imply the countability of rational numbers. As of now, it only includes Rational numbers on the interval $[-1,1]$. -Andrew Salmon 14:38, 11 September 2011 (CDT)


 * Good call. I suppose you can then create the countable union of all the intervals $[n .. n+1)$ for all $n \in \Z$ and establish that this handles all the rationals in $\Q$. Go for it, I've got other stuff I'm doing at the moment. --prime mover 14:42, 11 September 2011 (CDT)


 * Trivially fixed. Simply include the reciprocal of each of the numbers on the former (incorrect) list. ArcanaAwaits (talk)

Extra proofs
Both Schilling and Givant-Halmos give different proofs. How will we number them? &mdash; Lord_Farin (talk) 09:45, 29 March 2014 (UTC)


 * How about "Proof 1: Intuitive Proof" and the rest just Proof 2, 3, etc.? &mdash; Lord_Farin (talk) 10:23, 29 March 2014 (UTC)


 * I'm all for just naming them "Proof 1", "Proof 2", ... etc. The "intuitive proof" can be made rigorous by establishing that the ordering given can be constructed rigorously and captures every rational number in the sequence, so there is no reason for it to be "intuitive". --prime mover (talk) 20:12, 2 April 2014 (UTC)

Done as suggested. Please don't get wound up over the handwaving nature of Proof 4; that's how Schilling presents it, and I don't feel like putting in a lot of effort for a fourth proof. &mdash; Lord_Farin (talk) 20:50, 3 April 2014 (UTC)


 * No worries -- it's adequate. If we want to prove that the stated injection / bijection are in fact what they are said to be, then that can be added as and when someone feels like it, but I'm not too worried. Local entropy continues to reduce. --prime mover (talk) 21:05, 3 April 2014 (UTC)