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Forgive me if this is a novice question. I'm not a mathematics student, but I'm interested in mathematical philosophy.

Georg Cantor made an argument that the set of rational numbers is countable by showing a correspondence to the set of natural numbers. He did this by scanning rational numbers in a zigzag scheme starting at the top left corner of a 2D table of integers representing the numerator vs. denominator of every rational number. He also proved that the set of real numbers is uncountable through his famous diagonalization argument.

My question is, why can't real numbers also be counted in the same fashion by placing them in a 2D table of integers representing the whole vs. decimal parts of a real number i.e. like this:

 0 1 2 3 4 ... 0 0.0 0.1 0.2 0.3 0.4 ... 1 1.0 1.1 1.2 1.3 1.4 ... 2 2.0 2.1 2.2 2.3 2.4 ... 3 3.0 3.1 3.2 3.3 3.4 ... 4 4.0 4.1 4.2 4.3 4.4 ... . . . . . . . . . . . . . . . . . . . . .

and scanning them in a zigzag scheme starting at the top left corner? Negative reals can also be treated the same way as negative rationals (e.g. by pairing even natural numbers with positive real numbers, and odd natural numbers with negative real numbers).

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    $\begingroup$ Or even, for example, $0.12$. $\endgroup$ Commented Aug 2, 2013 at 10:55
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    $\begingroup$ Why can't the three digit numbers come later? And then the 4 digits one, etc. An infinite table... and then the infinite strings may come after the finite ones, like $\omega +1$ or something. $\endgroup$ Commented Aug 2, 2013 at 10:57
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    $\begingroup$ Aren't they going to show up in the table at some point? After all, it's an infinite table. $\endgroup$ Commented Aug 2, 2013 at 10:58
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    $\begingroup$ @EuYu But now you're just talking too rigorously about an informal concept, namely infinite table. The OP is trying to convey an idea non-rigorously (and I tried to help him with that), but you're talking about it on a different level of formalism. The way I see it there are two ways to go about this: the OP formalizes his infinite table and we go from there or we keep things simple and informal and we can't use what you just said. $\endgroup$ Commented Aug 2, 2013 at 11:08
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    $\begingroup$ @HaithamGad Yes, that's essentially the problem. The original table used by Cantor had an infinite number of columns (and rows), but the index of each column (and row) was finite. Right now you are considering a table in which the indices themselves need to be infinite. That's a whole different object altogether. $\endgroup$ Commented Aug 2, 2013 at 11:18

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You only counted a subset of the reals, namely, the set including the integers as well as reals with one decimal place. You cannot count the reals, as you would have to count to an infinite number of decimal places, as some reals have no fractional representation.

To 'count' as you propose, you would need the top heading to be reals, as well as the side heading, making it all but impossible to actually count anything, due to the fact you would now have two of the infinite sets, which you are using to count the infinite set itself.

Between any two real numbers lies the entire set of integers (Or so hypothesized. In any case, some infinite quantity), thus you cannot "count" reals, as there are infinite reals between any two reals you can pick to "count".

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  • $\begingroup$ WTH? Between any two real numbers lies the entire set of integers? Please tell me, what integers lie between the real numbers 0.3 and 0.7? $\endgroup$ Commented Jul 17, 2014 at 10:24

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