weakening naive comprehension to avoid the paradoxes 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?
 A: I'm not clear on why you don't regard ZFC as an example. You say: 


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. 


But it seems to me that the ZFC axioms of set theory result essentially from a weakening of naive comprehension, are highly popular, are well motivated and seem to avoid the paradoxes while having an abundance of understandable models. 
In particular, the ZFC axiom of separation is the result of weakening the naive comprehension axiom to the assertion that for any property $\phi$ and any set $A$, the collection $\{ \ x\ \mid\ x\in A\text{ and }\phi(x)\ \}$ forms a set. And one can similarly view the replacement axiom as an instance or weakening of naive comprehension, asserting of any set $A$ and property $\phi$ that the collection $\{\ x\ \mid\ \exists a\in A\ x\text{ is unique such that }\phi(x,a)\ \}$ forms a set.
Furthermore, the ZFC formulation of set theory seems to be very well motivated by the iterative conception of set, where one views the class of all sets being formed in a well-founded cumulative hierarchy formed in stages, in which the elements of a set are constructed at earlier stages than the set itself, and the stages continue in an endless transfinite progression. In essence, one must  construct the elements of a set before constructing the set itself.  On this philosophical view of how sets are formed, there is ample support for the separation and replacement axioms, and essentially none for the naive comprehension axiom (since there seems in general no reason to suppose all the $x$ with the desired property exist by some stage).
A: Ackermann set theory seems "pretty close" to having unrestricted comprehension, specifically in the form of the class and set comprehension schemas. Reinhardt proved that it is as strong as ZF (in particular, an additional replacement schema is not necessary). It's not entirely clear whether class comprehension is necessary.
I asked a question here about the strength of an even more minimalist fragment of Ackermann's set theory, which has only 1 or 2 axioms and a single comprehension schema. The form of the comprehension schema is, again, conceptually "quite close" to unrestricted comprehension.
