It is well known that Naive (to some, otherwise known as Ideal) set theory, that is, the set theory generated by the axioms:
(EXT) (x)(x $\in$ A iff x $\in$ B) iff A=B
(COMP) ($\exists$y)(x)(x $\in$ y iff $\phi$(x))
is inconsistent.
However, if one restricts $\phi$ to formulas using only conjunction and disjunction (as Thoralf Skolem does in his paper "Investigations On A Comprehension Axiom Without Negation In The Defining Propositional Functions", to use this as an example) one can show (as does he) that there exist domains (i.e. models) that satisfy both EXT and COMP i.e. that for these "positive propositions" (Skolem's term) Naive (Ideal) set theory is consistent. I will define the theory EXT + COMP for such $\phi$ that restriction to such $\phi$ which makes EXT + COMP consistent as a consistent fragment of Naive (Ideal) set theory.
My question is this: Has a maximal consistent fragment of Naive (Ideal) set theory been discovered yet? If so, can someone provide the citation or link to the paper(s) the result is contained in?