When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary.
Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely.
Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.
Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely.
Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.