2
$\begingroup$

I would like to show an idea on how to make real numbers infinitely countable. It is quite simple, too simple for me to believe it has been overlooked. So my question is, what have I overlooked?

enter image description here

So the "countable" list would start like this {0.1, 0.2, 1.1, 2.1, 1.2, 0.3, 0.4, 1.3, 2.2, 3.1 ...}, the way Cantor diagonally counted the list of rational numbers and potentially covering every real number there is.

And for the numbers after the decimal point, you go down a column and count normally after the decimal point but reverse the number.

$\endgroup$
1
  • 1
    $\begingroup$ It is really funny coming back to this question after 6 years and realizing how naive I was. Oh well. Progress through failure and humiliation. xkcd.com/1053 $\endgroup$
    – Xajoc8
    Commented Mar 2, 2021 at 13:07

2 Answers 2

Reset to default
3
$\begingroup$

This construction only covers the real numbers that have a finite decimal representation. These numbers are all rational - but you don't even cover all rational numbers, since they may have a periodic decimal representation [e.g. $\frac{1}{3} = 0.\overline{3}$].

$\endgroup$
3
  • $\begingroup$ But wouldn't 0.3333... occure in the list. After the decimal point the numbers are counted to infinity so 0.333333333... would just be at the end of the list? – Xajoc8 just now edit delete $\endgroup$
    – Xajoc8
    Commented Sep 10, 2015 at 7:08
  • $\begingroup$ No, there is no "end of the list". Exactly those numbers occur that have a finite place in the list. $\endgroup$
    – Dominik
    Commented Sep 10, 2015 at 10:29
  • $\begingroup$ Thank you, I appreciate your time and answer. There are clearly some things I haven't understood/heard of yet. Have a good day $\endgroup$
    – Xajoc8
    Commented Sep 10, 2015 at 17:53
1
$\begingroup$

All the numbers in your list have finite decimal expansions, so they are rational. In particular, $\sqrt 2$ is not on your list.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .