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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).

Hans Lundmark
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Haitham Gad
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  • So far the table only have decimal representations of rational numbers. – OR. Aug 02 '13 at 10:55
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    You are missing all real numbers which do not have a finite decimal expansion, your table does not even contain all rationals, for example $1/3 = 0.3333\ldots$. – martini Aug 02 '13 at 10:55
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    Or even, for example, $0.12$. – David Mitra Aug 02 '13 at 10:55
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    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. – Git Gud Aug 02 '13 at 10:57
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    Aren't they going to show up in the table at some point? After all, it's an infinite table. – Haitham Gad Aug 02 '13 at 10:58
  • @HaithamGad The key point is that it's a countably infinite table. Take a look at martini's example of $1/3$. When do you think that shows up? – EuYu Aug 02 '13 at 10:59
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    @EuYu Read my comment, please. – Git Gud Aug 02 '13 at 10:59
  • @GitGud I sort of understand the idea behind what you are proposing, but the purpose of the table is to demonstrate a bijection between $\mathbb{N}$ and $\mathbb{N}\times \mathbb{N}$. What is the purpose of your infinite extension? – EuYu Aug 02 '13 at 11:01
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    @EuYu It should show up in the zeroth row and the infinite'th column :) just as the 333333.../100000.... rational number. – Haitham Gad Aug 02 '13 at 11:01
  • @EuYu I suppose the OPs purpose is to find a bijection between $\Bbb R$ and $\Bbb N\times \Bbb N$. – Git Gud Aug 02 '13 at 11:04
  • There are probably much better ways to answer this, but basically, you would "run out" of natural numbers before you manage to finish. – rurouniwallace Aug 02 '13 at 11:04
  • Your table consists mostly of dots, so that's not even a foraml definition of a table. Can you specify explicitly which real number occurs in row $n$, column $m$? – Hagen von Eitzen Aug 02 '13 at 11:05
  • @GitGud The purpose of the diagonal argument is to demonstrate that no such bijection exists. Yes, you can consider the "infinite'th" column, but then you no longer have a map involving $\mathbb{N}\times \mathbb{N}$. – EuYu Aug 02 '13 at 11:05
  • @HagenvonEitzen n.m?! – Haitham Gad Aug 02 '13 at 11:05
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    @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. – Git Gud Aug 02 '13 at 11:08
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    @EuYu Why? Isn't 33333... a natural number? – Haitham Gad Aug 02 '13 at 11:09
  • @HaithamGad $333333\cdots$ is not a natural number. The set of natural numbers (i.e. $\mathbb{N}$) is infinite, but each given member of the set is finite. – EuYu Aug 02 '13 at 11:10
  • @EuYu Ok, now it's starting to make sense. So the problem boils down to not being able to "count" a number with infinite digits, right? – Haitham Gad Aug 02 '13 at 11:17
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    @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. – EuYu Aug 02 '13 at 11:18
  • @EuYu Got it. Thanks! (BTW, you can post this as an answer and I'll accept it). – Haitham Gad Aug 02 '13 at 11:21
  • @GitGud Because the argument was supposed to define a surjection of the naturals. In such a table one would have to use several times the naturals but then it is missing the argument about how many times the naturals were used. – OR. Aug 02 '13 at 11:33
  • BTW: If 3333.... is not a natural number, what's the right name for it? (is it transfinite?) – Haitham Gad Aug 02 '13 at 11:35
  • @HaithamGad: For the moment, until you define something you can do with it, it's just a string of digits. – hmakholm left over Monica Aug 02 '13 at 12:34
  • @HenningMakholm It's only hypothetically useful, just like the infinitely accurate $\pi$ or $\sqrt{2}$. – Haitham Gad Aug 02 '13 at 16:26
  • I'm not sure if you've come across this article, but it quite a long way for me when I was trying to understand un/countability: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument – MGA Jul 17 '14 at 09:56

1 Answers1

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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".

Afrenett
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  • 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? – gnasher729 Jul 17 '14 at 10:24