Comment by zkmon

I'm aware that I'm ignorant of many things, just like anyone else on this Earth. Some are less ignorant and some are more.

Could you be kind enough to explain the phrase "set of all integers" when the word all can not apply to an unbounded quantity? I think the word all is used loosely to extend it's meaning as used for finite sets, to a non-existent, unbounded set. For example, things such as all Americans, all particles in universe have a meaning because they have a boundary criterion. What is all integers?

I think one need to first define the realm of applicability or domain for the concepts such as comparison, 1-to-1 mapping, listing, diagonals, uniqueness, all etc.