If you define the natural numbers this way you'll run into Russell's paradox. For practical purposes that's fine and this is an elegant, modern restatement of F

If you define the natural numbers this way you'll run into Russell's paradox. For practical purposes that's fine and this is an elegant, modern restatement of Frege. But if you need to avoid Russell's paradox the sets defining each natural n can't contain n as an element. The easiest such construction was outlined elsewhere in this thread by xamael. Under your category theoretic construction, every natural number n is defined as the category of sets having cardinality n. But then every nth category will necessarily contain infinitely many sets with cardinality n that also contain n. That causes the paradox. Unfortunately I don't think you can construct the naturals in a category theoretic way while avoiding Russell's paradox since any category by cardinality will fall into that trap. But if you don't need to mind that problem, this is neat. Responding to your edit: I didn't downvote you; in fact I upvoted this comment because it's correct and it was gray at the time of my writing. My comment is just a point of clarification.