You can have the partial order A ≤ B iff A ⊆ B. But if you want size({1}) = size({2}), size({2}) = size({3}), etc., then you’ll find that size({1,2,3,…}) = size

You can have the partial order A ≤ B iff A ⊆ B. But if you want size({1}) = size({2}), size({2}) = size({3}), etc., then you’ll find that size({1,2,3,…}) = size({2,3,4,…}) and there’s really nothing you can do about it. But yes, the ordinals might be more to your liking. If you equip your sets with more structure you can say more things about them.