Russell's Paradox
Let denote the set R of all sets S that do not belong to themselves. The inconsistency of the existence of this set is known as Russell's paradox.
Solutions to the paradox restrict the notion of set construction in some way. To illustrate this in terms of our notation, let X = {x ∈ A : P(x)} denote the set of every element of A satisfying the predicate P(x). The canonical restriction on set builder notation asserts that X is a set only if A is already known to be a set. This restriction is codified in the axiom schema of separation present in standard axiomatic set theory. Note that this axiom schema excludes R from sethood.
Read more about this topic: Set-builder Notation
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