Cantor's diagonal argument works because it is based on a certain way of representing numbers. Is it obvious that it is not possible to represent real numbers in a different way, that would make it possible to count them?
Edit 1: Let me try to be clearer. When we read Cantor's argument, we can see that he represents a real number as an infinite sequence of binary digits. Using this representation, he shows that real numbers are uncountable. An intuitive counter argument could be that maybe there is another type (perhaps incredibly strange) way of representation that would make it possible to count the real numbers. A kind of trick, like the typical one used to show the countability of rational numbers. One could thus be tempted to think that when representing real numbers as infinite sequences of binary digits, it is those representations that are uncountable, but that some other representation could be countable. It seems to me that this can be summed up like this: Is a proof of the countability of a set dependent on the representation its members in the proof?
Best Answer
You are making the common mistake to confuse between a number and its decimal representation.
An easy way to see that $\Bbb R$ is uncountable, regardless to how we can or cannot represent real numbers, is to see there is an injection from $\mathcal P(\Bbb N)$ into $\Bbb R$, defined by $\displaystyle A\mapsto\sum_{n\in A}\frac1{3^{n+1}}$.
This function depends only on the fact that this sum is a convergent sum (as it is bounded by a convergent geometric sequence), and the fact that $\mathcal P(\Bbb N)$ is uncountable depends only on its property of being the power set of $\Bbb N$.