The opening sentence of Roger Myhill's article Paradoxes, in Synthese 60 (1984), 129-143, is: “Gödel said to me more than once "There never were any set-theoretic paradoxes, but the property-theoretic paradoxes are still unresolved"; and he may well have said the same thing in print.”
What, precisely, were Gödel's property theories. I have for a long time thought that they simply were set theories without the axiom of extensionality. But perhaps there is something more to them, e.g. something stemming from Gödel's ontological argument?