[jfarmer]

You're using too many imprecise terms. First, what does it mean for something to be able to be "unambiguously" described? As opposed to ambiguously described? You have to define it.

Second, what does it mean for something to be described (unambiguously or otherwise) with a "finite string?" What is a "string" here?

You're playing too loose with these ideas and it's biting you. You have to start by defining them precisely. For example, I don't see at all how the new number not on your list is "described unambiguously." It's presumably not enough to say "there is some number not on my list, we will call it x" since we know there is more than just one such number. How is that unambiguous?

In any case, that's why you have to define these things precisely.