A question on ‘the set of everything’. Alain Badiou - who takes set theory to be the best description of ontology that we have - makes a similar point but with an opposite conclusion. That the non-totalizibility of set theory attests not to any deficiency of set theory, but to the impossibility "totalizing" the universe at all. In other words, what you see as a deficiency in the instrument of description (set theory) is read by Badiou to be a positive characteristic of the world itself:
"In [Badiou's] language, the universe does not exist, whereas there are many worlds. [H]is argument for the nonexistence of the All or the universe draws on set-theoretical paradoxes, particularly Russell’s antinomy. Badiou argues as follows: If the All existed, it would have to exist as a member of itself. Otherwise, there would be an all outside of which something else, namely the All, existed. Hence, the All has to be a member of itself. Thus, there is at least one set, which is a member of itself. Nevertheless, there are obviously sets that are not members of themselves.
The set of all bananas is not itself a banana. This entails that the All consists both of sets which are members of themselves and sets which are not members of themselves. Given that the set of all sets that are not members of themselves famously leads into Russell’s paradox, the All cannot exist, because its existence would entail an antinomy". (Markus Gabriel, Transcendental Ontology).
I don't like the idea of treating set theory as ontology at all, so it's a non-starter for me, but I thought this was interesting.