Is naive set theory, simply set theory that has been left unformalised as the entry in Wikipedia suggests?
However, the SEP, in its entry on inconsistent mathematics, suggests that:
It should also be noted that Brady's construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is reducible to logic via the naive comprehension schema. The only change needed is a move to an inconsistency-tolerant logic.
One should note here that the denial of Explosion, allows the assumption of Russells set - the set of all sets that are not members of themselves, and the universal set - the set of all sets without leading to paradox.
Does naive set theory when assuming a paraconsistent logic:
a. Has a simpler axiomatisation than ZFC?
b. What is the status of Choice in Naive Set Theory
c. Can successfully push through Freges Logicist programme?