Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because no consistent weakenings seem to be particularly well motivated or even to lead to understandable models. At any rate, that is so of the most famous consistent (well, *probably* consistent) weakening, New Foundations. Nonetheless, it could be illuminating to understand the partial order of consistent subtheories of naive set theory. My question would be: what is known about it? Unfortunately, however, that question seems ill-posed in that an arbitrary axiom $\psi$ can be coded as comprehension for $(\psi \land x\neq x)\lor (\lnot\psi \land x\notin x)$. But can anything interesting be said?

Besides NF, I am aware of just one type of set theory that is (almost) naturally thought of as arrived at by admitting only a subset of all possible instances of naive comprehension. These are the so-called positive set theories. Unfortunately, I know nothing about them except that they admit a universal set (like NF), and they apparently do require some extra axioms that are not naturally expressed as instances of comprehension. In particular, according to Wikipedia, the theory known as $GPK^{+}_{\infty}$ requires the axiom of infinity, the empty set axiom (!), and an axiom scheme of "closure" giving, for each formula $\phi$ with one free variable, the intersection of all the sets that contain every $x$ such that $\phi(x)$. This seems to me arguably in the spirit of restricting naive comprehension because comprehension is still the main set construction principle, and in particular there is no need for powerset or replacement. Are there other "natural" examples of set theories that can be thought of as arrived at by weakening naive comprehension? Perhaps ones that *don't* admit a universal set?

Even non-effective examples (examples where the set of instances of comprehension that is admitted is not computably enumerable) might be interesting. Also, there might be interesting ways of weakening naive comprehension that are different from simply restricting the allowed instances of the schema. For instance, maybe some set of disjunctions of instances of naive comprehension is interesting, or maybe it is interesting to consider an axiom that would only guarantee the existence of a set that is in some sense "close" to the class of objects satisfying $\phi$.

The context in which this question came up is that I was trying to explain Russell's paradox to someone, and their reaction was, well you should just throw out the instance of the comprehension axiom that leads to paradox. Of course, throwing out literally that one instance won't restore consistency. But pointing out a flaw in any particular proposal someone with this attitude toward the paradoxes might propose wouldn't show that some more sophisticated proposal might succeed. I was hoping for some sort of general argument that, say, proceeding in this way inevitably leads to a system that is either like NF or like positive set theory (if it is not inconsistent or extremely weak). (What else could be wrong with $x\notin x$ except that it is unstratified or that it involves negation?) Or at least an argument that you won't get an extension of ZF by any natural weakening procedure would be nice! Both NF and positive set theory, if I understand the situation aright, could serve as a foundation for mathematics, but both are less intuitive and convenient than ZFC, and it is sort of an article of faith for set theorists that any alternative to ZFC we might ever find is either somehow worse than ZFC or not deeply different from it, yes?

consistent(as opposed to paraconsistent). $\endgroup$ – Mike Shulman Sep 14 '17 at 8:33