There are two classical paradoxes associated with set theory; and that is the existence of the Universal set and the Russell set.
The usual set theory takes the notion of element as basic; these are the atoms by which sets are built.
In the debate between Aristotle and the atomists; Aristotle took the tack the universal divisibility isn't possible ie atoms do not exist.
Could one take a similar tack with set theory - that this is possible as a coherent strategy is shown by the existence of Category Theory ie there are categories without global elements.
That is start from the supposition that there is a universal set; and see what can be drawn from this?
Ie a set-theory from a top-down perspective rather than a bottom-up one.
(If such a theory could be constructed; it would be ironical, and apposite (or perhaps, opposite) if elements were shown to be inconsistent).