# Positive set theory, antifoundation, and the “co-Russell set”

A more focused version of this question has now been asked at MO.

Tl;dr version: are there "reasonable" theories which prove/disprove "the set of all sets containing themselves, contains itself"?

Inspired by this question, I'd like to ask a question which has been vaguely on my mind for a while but which I've never looked into.

Working naively for a moment, let $$S=\{x: x\in x\}$$ be the "dual" to Russell's paradoxical set $$R$$. There does not seem to be an immediate argument showing that $$S$$ is or is not an element of itself, nicely paralleling the fact that there are of course arguments for $$R$$ both containing and not containing itself (that's exactly what the paradox is, of course).

However, it's a bit premature to leap to the conclusion that there actually are no such arguments. Namely, if we look at the Godel situation, we see something quite different: while the Godel sentence "I am unprovable (in $$T$$)" is neither provable nor disprovable (in $$T$$), the sentence "I am provable (in $$T$$)" is provable (in $$T$$) (as long as we express "is provable" in a reasonable way)! So a certain intuitive symmetry is broken. So this raises the possibility that the question

$$\mbox{Does S contain itself?}$$

could actually be answered, at least from "reasonable" axioms.

Now ZFC does answer it, albeit in a trivial way: in ZFC we have $$S=\emptyset$$. So ideally we're looking for a set theory which allows sets containing themselves, so that $$S$$ is nontrivial. Also, to keep the parallel with Russell's paradox, a set theory more closely resembling naive comprehension is a reasonable thing to desire.

All of this suggests looking at some positive set theory - which proves that $$S$$ exists, since "$$x\in x$$" is a positive formula, but is not susceptible to Russell's paradox since "$$x\not\in x$$" is not a positive formula - possibly augmented by some kind of antifoundation axiom.

To be specific:

Is there a "natural" positive set theory (e.g. $$GPK_\infty^+$$), or extension of such by a "natural" antifoundation axiom (e.g. Boffa's), which decides whether $$S\in S$$?

In general, I'm interested in the status of "$$S\in S$$" in positive set theories. I'm especially excited by those which prove $$S\in S$$; note that these would have to prove the existence of sets containing themselves, since otherwise $$S=\emptyset\not\in S$$.

• @egreg I'm aware. Note that positive set theory does prove that $S$ is a set, as I said in the paragraph before "to be specific." – Noah Schweber Sep 15 '17 at 23:15
• @egreg Actually, ZFC does let us define $S$ (note that it can also define the class of sets not containing themselves) - even better it proves that $S$ is a set, since it proves that $S=\emptyset$. – Noah Schweber Sep 15 '17 at 23:16
• Well, $x\in x$ is not stratified, so NF is not gonna prove it is a set. – Asaf Karagila Sep 15 '17 at 23:17
• @AsafKaragila NF is not positive set theory. I didn't mention NF. – Noah Schweber Sep 15 '17 at 23:17
• The 'standard model' of positive set theory $+\neg \mathrm{Inf}$ has a natural topology that is a compact computable metric space. In particular the set membership relation is co-c.e. between 'computable sets' (sets that are computable points in this metric) and I'm fairly certain $S$ is such a computable point. So if the answer is no for this model a(n extremely slow) computer search can tell us, but my gut tells me $S\in S$ in this structure. – James Hanson Jun 17 '19 at 7:04

This is not a direct answer to your specific question, but it might shed an idea on a possible solution within the arena of $$\mathsf{GPK}_\infty^+$$ in which your question is decidable and to the positive!
Around three months ago I've asked Olivier Esser if whether adding the following condition is consistent with $$\mathsf{GPK}_\infty^+$$:
$$$$ if $$\phi$$ is purely positive without free variables other than $$y,A$$, and without using the false formula, then: $$\exists A \forall y (y \in A \iff \phi)"$$ By this principle we can construct Quine atoms and alike sets, which are not constructible in merely $$\mathsf{GPK}_\infty^+$$