In the context of set theory, by "self-reference" I would assume you mean either 1) impredicative definitions, i.e., definitions where the bound variables used

In the context of set theory, by "self-reference" I would assume you mean either 1) impredicative definitions, i.e., definitions where the bound variables used in the definition are permitted to range over the whole universe to which the very thing being defined belongs, or 2) non-well-founded sets, sets that can have themselves as members or be defined in terms of infinitely descending membership chains like ...ϵ n4 ϵ n3 ϵ n2 ϵ n. ZFC admits the first by not limiting the range of the bound variables of the axiom schemas of Separation or Replacement (unlike the Separation axiom schema of Bounded Zermelo Set Theory), but it disallows the second (due, for example, to Foundation). So at least one form of self-reference, impredicative definitions, are permitted by ZFC.