As with introducing Equivalent Sets, we want to talk about the size of a set.1
A finite set is one that is equivalent to some natural number; an infinite set is one which is not finite. From this we can show that $\omega $ is infinite. This justifies the language “principle of infinity” with Natural Numbers. The principle of infinity asserts the existence of a particular infinite set; namely $\omega $.
It happens that if a set is equivalent to a natural number, it is equivalent to only one natural number.
A consequence is that a finite set is never equivalent to a proper subset of itself. So long as we are considering finite sets, a piece (subset) is always less than than the whole (original set).
Every subset of a natural number is equivalent to a natural number.3 A consequence is: