This isn't iconoclasm, it's ignorance.
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.