For each formula , the axiom schema of specification forms the subsetof any set . When interpreting the schema inside a class model, the formula must be evaluated internally.
A predicate selects a subset of a previously given set . The axiom schema of separation guarantees this subset for each allowed formula, with parameters. Unlike unrestricted universal set formation, the resulting set has an ambient bound. Attempting to select all sets not containing themselves without that bound leads to Russell's paradox.
Let be a transitive class. If is closed under the ambient subsets defined by , then satisfies the -instance of separation, because
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The Axiom Schema of Specification (also known as the Axiom Schema of Separation) is a fundamental principle in set theory, particularly in the context of Zermelo-Fraenkel set theory (ZF). It is one of the axioms that govern how sets can be constructed and manipulated within this framework. In essence, the Axiom Schema of Specification allows for the creation of a new set by specifying a property that its elements must satisfy.