Ok. So are you suggesting a notion of "size" such that A is a smaller "size" than B if and only if A is extensionally equal to a proper subset of B? This would

Ok. So are you suggesting a notion of "size" such that A is a smaller "size" than B if and only if A is extensionally equal to a proper subset of B? This would lead to a partial order on all sets. As a result, the perfect squares would smaller than the naturals, but also it would make many sets incomparable in size, including the natural numbers and the set {A}. Or if this is not what you are suggesting, what are you suggesting?