Russell Paradox and set theories

Question

The Russell paradox arise in the Cantor set theory, but it can be avoided in the $ZF$ and in $NGB$ axiomatic set theory. Are there other axiomatic set theories in which this paradox can be avoided? Thanks.

Answer 1

As it is mentioned in comments above, the so called Russell paradox is a consequence of non-restricted Comprehension Schema according to which for any formula $\varphi(x)$, where $x$ is free, $\{x\mid\varphi(x)\}$ is a set. This paradox is actually a result of a logical truth ($R$ is a binary predicate): $$ \neg\exists x\forall y(yRx\iff \neg yRy)\,. $$ In light of this, assuming non-restricted comprehension in a language in which you have at least one binary predicate you always get inconsistent theory. Thus if you want to build a set theory based on classical logic you must restrict the schema one way or another.

EDIT: This address Asaf question below (I should have written it before it was asked). As I wrote above, if you build a system of set theory you must restrict the Comprehension Schema. All such restrictions I am aware of allow you to avoid falling into inconsistency due to Russell paradox.