There is the intuition that if one set is a proper subset of another, the former is smaller than the latter. For example, there are "more" numbers in the set of natural numbers $\mathbb{N}$ than in the set of even numbers = $\{0,2,4,\dots\}$.
The part-whole relation is a partial order. So the problem is how to extend it "naturally".
Can we give a criterion of ‘size’ based on this concept?
A first step may be the power set of the set of natural numbers. For example, we would like the set $\{the \ even \ numbers\}\cup\{1\}$ and the set $\{the \ even \ numbers\}\cup\{2\}$ be the same "size". A first try may be to assign each subset of natural numbers a "size" by its natural density. However, this will make all finite subsets of natural numbers, and other subsets like the set of the primes the same "size".