Axiom schema of specification

The axiom schema of specification asserts that, for any set and formula, the elements of that set satisfying the formula form a set. It restricts set formation to subsets of existing sets.

Predicate logic: A formal system for expressing statements about objects, their properties, and relations. Each formula supplies the condition that selects elements from an existing set.
Ernst Zermelo: A German mathematician who formulated an influential axiomatization of set theory. His 1908 axioms included the principle later called specification.
Set theory: The mathematical study of sets, membership, and the structures built from them. Specification is a foundational principle governing set existence.
Axiom schema of replacement: A schema asserting that the image of a set under a definable function is a set. Replacement supports set formation by definable images, rather than merely selecting elements.
Subset: A set whose every element belongs to another set. Specification guarantees that the defined set is a subset of the original set.
Naive set theory: An informal approach to sets that permits formation from intuitive descriptions. Its unrestricted set formation exposed the paradoxes axiomatic set theory addressed.
Set (mathematics): A collection treated as a single mathematical object, with membership as its basic relation. The schema starts with an already existing set whose elements can be selected.
Axiom of foundation: An axiom asserting that every nonempty set contains an element disjoint from it. Foundation constrains membership structure, whereas specification constrains subset formation.
Comprehension axiom: An unrestricted set-formation principle that creates a set from every condition. Unrestricted comprehension is the paradox-producing principle that specification restricts.
Zermelo set theory: An axiomatic set theory introduced by Ernst Zermelo in 1908. Specification was among the principles in Zermelo's original axiomatization.

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