Is the set of all digits in a real number countable/listable? It seems to me that it is, based merely on my layman's reading of popular books about set theory. In other words, it seems that one can map each position in a real number to one of the natural numbers.
But as usually happens with math, I am probably missing something.
Thanks for any help.
Question edited based on the answer from fleablood. I think he has properly paraphrased my confusion on this topic.
"Perhaps you are thinking that if sequence that makes up the real number is countable, that that should mean the number of real numbers (or equivalently the number of possible countable sequences) should also be countable. But you know that there is a rule written in stone 'the Real Numbers are Uncountable' and you are wondering where your error is.
It is not the case that the number of countable sequences should be countable. Just there are an infinite number of finite numbers, there are an uncountable number of countable sequences."
Yes, that was my thinking almost exactly. I know the real numbers have been proven to be uncountable, but I have trouble understanding how.
For example, the set of digits in real number r1 is countable. The set of digits in real number r2 is also countable. If you add them, r1 + r2, you get real number r3 whose set of digits is also countable. I imagine a never-ending series such as r1 + r2 + r3 ... rN. If that sum is always a real number with a countable set of digits I don't understand where along the line the result is an uncountable set.
My guess is that I am begging the question somehow (r1 + r2 + r3 ... rN doesn't say anything about the size of the set being used for the sum) but it still nags me.
Is there a layman's version of the proof for "there are an uncountable number of countable sequences"?