Definition
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[1Y0]The axiom of specification, which reads
If \(A\) is a set, and \(P(x)\) is a logical proposition, then \(\{ x∈ A:P(x)\} \) is a set.
Formally, setting \(B=\{ x∈ A:P(x)\} \),
\[ \forall X, X\in B \iff X\in A\land P(x)\quad . \]
This axiom avoids Russell’s paradox: let \(A\) be the set of \(x\) such that \(x∉ x\), then you have neither \(A∈ A\) nor \(A∉ A\).