# Is there a couniversal solution to the predicate-theoretic version of Russell's paradox?

In set theory, let us call a solution to the problem of universal-sets-or-proper-classes a couniversal solution when it involves proposing the following:

1. Ux((xU) ⟺ (xU))

This means that U is a set of all other things (sets, there); but so now the reason why this pertains to Russell's paradox is that we should like to speak of a well-founded set of all other well-founded sets. Or, then, a noncircular set of all other noncircular sets, which does not trigger the paradox (or so it seems to me).

The SEP article on self-reference mentions a semantic paradox akin to the set-theoretic one, and I had thought that either the article on Russell's paradox specifically, or the broader article on paradoxes relative to contemporary logic, went over the formulation of Russell's paradox in predicate-theoretic form, but I'm having a lot of trouble finding the exact section of the exact article that I have in mind, here. At any rate, let's use, "Is 'heterological' a heterological term?" as our equivalent. (Correct me if I'm wrong about that!) Is there a way to formulate the counterpart of a couniversal set, in predicate-theoretic (modulo set-theoretic) terms, such that a couniversal predicate(?) deflects the other form of Russell's paradox?

• Are you aware that Russel's Paradox is not the only problem with a universal set? Another problem is that the powerset of the universal set is bigger than the universal set. Commented Oct 19, 2023 at 15:34
• @DavidGudeman both Quine's set theory NF, and Olivier Esser's positive set theory, have universal sets. The easiest way to do that modulo the powerset axiom is to just deny the powerset axiom on the universe. There is no proof that every set has a powerset anyway, hence why it's an axiom. Commented Oct 19, 2023 at 15:51
• I should add that, to my knowledge, in the NF (or maybe NFU, I don't recall ATM) world, the powerset of the universal set is smaller than the universal set, which is very weird but I think has something to do with the universal set having elements that aren't able to be collected into subsets, so the number of elements the universe has altogether is greater than the number of elements that can be collected into subsets, and then the collection of those subsets is what is somehow smaller. I'll check back on the details but I'm pretty sure that's more or less how it goes, there. Commented Oct 19, 2023 at 16:15
• For your formulaic spec there's already a famous big almost universal set aka Grothendieck universe consisting of small sets in the sense of category theory avoiding Russell like semantic paradoxes along with proper classes in algebraic geometry, which is compatible with well-foundedness and itself is some member of an even larger Grothendieck universe ad infinitum, forming some towered multiverse metaphorically. Of course non well-founded NF or positive set theory (PST) with naive comprehension also work. It's bit pity there's not much interest in PST or its opposite as amorphous sets... Commented Oct 19, 2023 at 22:49
• 'heterological' seems like a negative predicate thus 'heterological' is a heterological term cannot be well formed in PST metaphorically speaking but there's a positive set of all positive sets in a naive manner... Commented Oct 20, 2023 at 0:12