Full Idea
Class Nominalism cannot explain co-extensive properties (which qualify the same things), and also a random (non-natural) set has particulars with nothing in common, thus failing to capture an essential feature of a general property.
Gist of Idea
'Class Nominalism' cannot explain co-extensive properties, or sets with random members
Source
David M. Armstrong (Universals [1995], p.503)
Book Reference
'A Companion to Metaphysics', ed/tr. Kim,Jaegwon/Sosa,Ernest [Blackwell 1995], p.503
A Reaction
These objections strike me as conclusive, since we can assign things to a set quite arbitrarily, so membership of a set may signify no shared property at all (except, say, 'owned by me', which is hardly a property).