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If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive integers, and then listing those in an array like this: 

1/1 2/1 3/1 4/1 ...
1/2 2/2 3/2 4/2 ...
1/3 2/3 3/3 4/3 ...
....

We know that each row can go on for infinity, so instead we can draw a diagonal lines through the array to ensure that every fraction is counted, right?

But if we take all real numbers and try and list or count them, allegedly we can't, but why?

For example, if we wanted to list or make countable all numbers between 0 and 1, why can't we say that all possible permutations of decimal number combinations could be listed or counted the same way that all fractions are countable or listable for positive rational numbers?

If we're unable to list or count all possible combinations of decimal numbers between 0 and 1, why are we not saying that all the infinite combinations of fractions used to make up all positive rational numbers isn't countable as well?

I realize that a very erudite and technical explanation of the Cantor Diagonalization Argument was provided in this post, however I'm not specifically asking for an explanation of this argument and moreover the very thorough explanation provided by Arturo Magidin for the OP in the post is slightly beyond my understanding as I'm very new to set theory and discrete mathematics.

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AdjunctProfessorFalcon
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    You need to look up Cantor's diagonal argument. – Gregory Grant Feb 10 '16 at 22:25
  • Briefly, because.. Cantor's Theorem (the diagonal argument). Given any enumeration of reals (function $\Bbb N\to\Bbb R$), the theorem constructs a real that is not enumerated (it isn't in the range of the function). This proves the result, right? There's a big difference (in size) between the set of finite strings of digits, and the set of infinite strings of digits. – BrianO Feb 10 '16 at 22:27
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    In part, it's because all rationals can be described in a finite number of symbols (ex. 13/19) but some reals need an infinite number of symbols (ex. 2.816452…). But look at Cantor's argument for an actual proof. – Akiva Weinberger Feb 10 '16 at 22:40
  • @AkivaWeinberger So there is actually a finite number of fractions for each rational but a potentially infinite number of decimal combinations that are between 1 and 0, so simplistically we never "get" to 1 because we can never list all the decimal combinations between 0 and 1? – AdjunctProfessorFalcon Feb 10 '16 at 22:42
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    @Malvin9000 What does "there is actually a finite number of fractions for each rational" mean? For the rational $0.5$, there are infinitely many fractions ($\frac12$, $\frac24$, $\frac36$, etc.), for example. But there are finitely many symbols per each fraction, and so we only need finitely many symbols to uniquely name a rational. – Akiva Weinberger Feb 10 '16 at 23:30
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    @Malvin9000 If the reals were countable, we would be able to make a list of them, and name each real by where it appears in our list. We could call them "the first one in our list," "the second one in our list," etc. But that contradicts the idea (that I haven't actually proven) that some reals would need infinitely many symbols to name. – Akiva Weinberger Feb 10 '16 at 23:32
  • @AkivaWeinberger What's the difference between fractions and symbols in this context? – AdjunctProfessorFalcon Feb 10 '16 at 23:33
  • @Malvin9000 I interpreted "fraction" to mean a collection of digits, a slash, and then more digits, such as "3/6." A symbol would be something like a digit, or a letter, or a slash, or something like that. So "3/6" is three symbols. – Akiva Weinberger Feb 10 '16 at 23:40
  • @AkivaWeinberger Ok, just wanted to be sure I understood what you meant — in other words, we can come up with a potentially infinite list of fractions to represent say 0.5, but all of those fractions mean the same thing (represent 0.5) . But we can't ever come up with a finite list or set of all decimal number combinations between 0 and 1, right? – AdjunctProfessorFalcon Feb 10 '16 at 23:59
  • @Malvin9000 You know what, just look at Cantor's argument. – Akiva Weinberger Feb 11 '16 at 00:36
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    @AkivaWeinberger Sorry to tax your patience buddy, if I understood Cantor's argument I wouldn't have asked this layman's question in the first place, right? – AdjunctProfessorFalcon Feb 11 '16 at 00:44

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