>> Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of even a single irrational number in your system.

That is my point exactly. Why do we accept that there are real numbers that cannot be defined by a finite string of symbols, but there are no integers of that nature.

Getting back to the axiom of choice thing. How do you pick a specific item from one of those ill-defined sets that include numbers that don't have finite descriptions? How do you choose something you can't name? It's absurdity all the way down. ;-)

That did not seem in any way to be your point. You said

> I've never been convinced that there are more reals than integers

I'm explaining that your method of mapping reals to integers is flawed, and so does not prove that reals and integers have the same cardinality.

> Why do we accept that there are real numbers that cannot be defined by a finite string of symbols

We don't accept it. It's easy to prove that it's the case.

I think you are unfamiliar with how R is defined. The traditional definition of R is a completion of Q with respect to the metric |x-y| (getting all Cauchy sequences to converge). That's larger than infinite sequence of decimals.