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- Defeating Russell's paradox 10 answers
For a collection $C$ define an object $b \in C$ iff $b$ is a class and $b \notin b$ and otherwise $b \in C$. Then we can decide any object whether it belongs to $C$. In Hungerford's Algebra p.2 it states in illustrating Gödel-Bernays axiomatic set theory that
Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or element) of $A$
Since it's possible to determine whether an object belongs to $C$, $C$ must be a class.
Then there is a paradox whether $C \in C$. How to avoid this in Gödel-Bernays axiomatic set theory.