Thus, with partitioning into types, the notion of self-membership, along with non-self-membership, could not even be deployed.
However, there seems to have been always a desire to create a theory that allows self-membership or self-application.
So self-membership does not make sense, the universal set does not exist and Russell's paradox disappears.
Zermelo's view of sets as axiomatized in 1908 did not in itself exclude the possibility of self-membership.
For that requires that the predicate x ∈ x of self-membership be meaningful (and well-formed); which it is not.
The point to note is that Russell's paradox and the liar paradox depend crucially on circular notions (self-membership and self-reference).
The ramified theory of types due to Whitehead and Russell, and ZF (the formalization of set theory named after Zermelo and Fraenkel) exclude self-membership.
Cantor's and Russell's paradoxes that were discovered in the late 19th early 20th century both involve self-membership of a set.
Dialetheism also affords a treatment of the set-theoretic paradoxes via set theories based on an unrestricted "comprehension schema" for sets: for any condition or property, including paradoxical ones like non-self-membership, there exists a corresponding set.
Types are intrinsic to logical and mathematical objects and the logical paradoxes are exactly those which require type distinctions to be solved (e.g., self-membership is blocked for objects in the type-theoretic hierarchy).
Finally, and more generally, it may be observed that the possibility of mereological loops is to be taken seriously for the same sort of reasons that led to the development of non-well-founded set theory, i.e., set theory tolerating cases of self-membership and, more generally, of membership circularities (Aczel 1988; Barwise and Moss 1996).
