I was thinking about the ontological problem of Universals versus Particulars and there is one thing that I can't understand well about the concept of Universal. If I have a cat (let's call it Steve) I understand that Steve is an Individual but shares several properties with other cats, among which there is also the "Catness". Now "Catness" is a universals, the pure ontological concept of "cat".
What I am trying to understand without result is how such a thing can exist. When our mind thinks about the concept of cat, thought is inevitably accompanied by a shape and a color and other properties. However by definition the universal catness must be devoid of shape or color, otherwise it would be reduced to a Particular.
When I think of this problem, it occurs to me that universals are probably sets while individuals are the elements of sets, for example in the case of cats if we take C ={c1, c2, c3, c4, c5, c6.....} as the set of all cats then C is the universals and c1, c2, c3... are all individuals. From this reasoning it is derived that individuals are only the intersections between sets (Universal).
Is my reasoning right?