If I understand number theory correctly, a set of numbers can be infinite but not exhaustive.
For example, I could indicate "all the integers which end in the digit 5" or something. And it would be infinite, as it would go on and on and on and on and on and on and on .... yet not exhaustive, as the set indicated most certainly does not contain ALL numbers.
For some reason I couldn't find a good link to back this up, but I can look tomorrow. I need to log off too.
OK, but numbers by their very nature are symbolic, they are not natural. Symbols represent, they are not the thing unto themselves.
That's where a lot of folks get lost on a tangent. We are so ate up in our minds with symbols that we've forgotten that symbols are just that...symbols. Of themselves, symbols are meaningless. I can't eat a letter "A" or a number "3" or a "$" sign, they are NOT real.
Humanity has developed a symbolic system that helps us parse information, and that began in earnest during the Agricultural Revolution, +/- 10,000 bc, the end of the last Ice Age and beginning of walled cities as people started sticking to one spot to tend a crop. Grain is not people food, humans did not evolve to consume grain. (I've gone over this many times in past essays, before your time.)
Think to when you were a child, before you learned an alphabet or numbers. How did you think? How did you reason?
But I was just a child...OK, but it doesn't change the fact that in the absence of symbolic thought, our minds think in pictures. This is closely related to imagination, so people still do it and don't realize.
We are so focused on the menu, we forget to order and enjoy the meal. We "adults" are so consumed by symbolic thought, we forget that symbols are not real, they only point to reality (in a best case scenario).
Symbolically representing infinity will always be meaningless in reality. Infinity in reality is ALL, there is nothing to add, and nowhere to set a detraction aside. Symbolically all kinds of clever articulations can be done, but symbols are not and never were "real."