Material set theory

Specification Axiom Schema

The Specification Axiom, technically an axiom schema, is a possible axiom schema of Material set theory: zf Let 𝜑(𝑥) be a Predicative formula (i.e. unary definite condition) of set theory. Then,

(∀𝔐⁡𝐴)(∃𝔐⁡𝐵)[𝑥∈𝐵⟺𝑥∈𝐴∧𝜑(𝑥)]

which is to say, there exists a subset 𝐵 of 𝐴 containing all those elements 𝑥 of 𝐴 for which 𝜑(𝑥). By the Axiom of Extensionality such a set is unique, and we denote it using Set builder notation as

𝐵={𝑥∈𝐴:𝜑(𝑥)}

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