Proof that the real numbers are countable: Help with why this is wrong
I know there are uncountably many real numbers between 0 and 1, and I am trying to figure out why this argument is wrong:
Proof that there are finitely many real numbers between 0 and 1:
Consider the string of digits after the decimal point.
We start with strings of length 0, then length 1, then length 2.
Each set of strings has only finitely many elements, so they can be numbered.
We will eventually reach every string by this process.
Therefore we can enumerate all the real numbers between 0 and 1.