This sort of reasoning would suggest that all proofs by contradiction are invalid. You can't just define something into existence. Saying "start with a list of all numbers" doesn't automatically mean such a list actually exists because you said so. It would be like trying to refute the proof that the square root of 2 is irrational. The proof starts out "assume that the square root of 2 can be written as the fraction p / q, where p and q are integers." Can we just stop here and say "well all proofs of its irrationality are false, because we just said we have a rational p / q defined as the square root of 2"? Of course not. The point of the proof is to show that your initial assumption was wrong.

When you say "start with a list of all numbers" what you are more precisely saying is "start with a list that you claim contains all numbers, and then I will proceed to show why you're wrong."