- If mereological universalism holds, then for any predicate F that is satisfied by an object, there is a mereological sum of all the objects that satisfy F.
- If Platonism is true, then for any object x there is a property Ux such that (a) x has Ux, (b) nothing else has Ux, and (c) Ux does not have mereological parts.
Assume Platonism and mereological universalism. For any predicate G(x), say that IG(P) if and only if P is an identity of an object x such that G(x). For any object S, say that x∈S if and only if an identity of x is a mereological part of S. Then by (1) and (2), for any predicate G that has at least one satisfier there is an object M such that x∈M if and only if G(x). Just let M be any mereological sum of all the objects satisfying IG (to get that if x∈M, then G(x), note that it follows from (c) that no identity can overlap more than one identity).
But now we have a contradiction. Let G(x) be the Russell predicate not(x∈x). Then G has at least one satisfier: e.g., you satisfy G. So there is an M such that x∈M if and only if G(x). So, M∈M if and only if not(M∈M), a contradiction.
So, our Platonist mereological universalist needs to restrict her universalism to concrete objects or something like that. Lewis's Parts of Classes is no doubt relevant, and what I said above may well overlap with Lewis (I haven't actually read that book--just ordered from Interlibrary Loan).