You are confused about the concept of countability and finiteness. The enumeration that you mention requires that a set be countable, not that it is finite. So

You are confused about the concept of countability and finiteness. The enumeration that you mention requires that a set be countable, not that it is finite. So it's wrong to somehow require the enumeration to finish. Countable sets can be finite or infinite. Most interesting countable sets are infinite. The enumeration then means a bijection with the natural numbers. Your definition of enumeration seems to require it to "complete" which is incorrect. (I put "complete" in quotes because that's the word you used, but I don't want you to further confuse that with the completeness of real numbers which is a whole another thing.)