> If there exists a procedure to describe which number to choose (i.e. an algorithm/formula), you don't need AC at all.
The problem here is not that there does not exist a procedure to describe which number to choose, the problem is that there does not exist any way to describe which number you chose. Those are not the same thing.
I'm not sure I understand what you are asking. One is a description of a procedure and the other is a description of a number so obviously these are two different things.
The problem here is not that there does not exist a procedure to describe which number to choose, the problem is that there does not exist any way to describe which number you chose. Those are not the same thing.