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.
Russell Paradox and set theories
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2@Mathemagician1234: I'm not certain if the CS restriction is _more_ important in either NBG or MK set theory than in ZFC. In either case, it serves the same purpose. A ZFC-like theory with an unrestricted CS would give a _set_ of all sets, since there is nothing else for this definable object to be. NBG has the the Axiom of Limitation of Size which itself implies that $V$ is a proper class. – 2012-03-21
1 Answers
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.
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0@Asaf: I edited my answer a bit. Hope you can read my intentions now. – 2012-08-31