[qsort]

You're probably trying to say something but I don't understand what.

The way you 'have' infinity in math is by postulating there exists an inductive set (axiom of infinity in both ZF and NBG), and constructing other infinite sets using that as a building block.

Your first point is a definition of an infinite set (there are a bunch of equivalent ones), your second point is a statement of the axiom of infinity I assume?