Axiom schema of replacement

The axiom schema of replacement asserts that the image of any set under a definable functional relation is a set. It supports transfinite constructions whose outputs may not be bounded by an existing set.

Functional formula: A formula that assigns exactly one value to each input in its specified domain. Replacement applies when the defining formula gives a unique output for every member of a set.
First-order logic: A formal logic with variables for individual objects and quantifiers over a specified domain. Each instance of replacement is expressed as a first-order axiom in the language of set theory.
Cumulative hierarchy: The staged universe of sets built by iterating the power-set operation along the ordinals. Replacement helps ensure that stages indexed by a set of ordinals can be gathered into a set.
Axiom of collection: A schema asserting that witnesses for a definable relation on a set can be gathered into a set. Collection does not require unique outputs; in standard set theory it is closely related to replacement.
Image of a function: The set of outputs a function takes on its domain. Replacement guarantees that the outputs assigned to a set of inputs form a set.
Set: A collection treated as a single mathematical object in set theory. Replacement begins with a set of inputs and asserts that their defined outputs also form a set.
Ordinal arithmetic: Arithmetic operations on ordinals, the order types of well-ordered sets. Replacement supports definitions that assign ordinal values across set-sized domains.
Axiom of boundedness: An axiom asserting that every set is contained in some level of the cumulative hierarchy. In ZF, replacement yields boundedness, expressing a consequence of its set-collection strength.
Transfinite recursion: A method for defining objects successively along an ordinal, using earlier stages to specify later ones. Replacement collects the stages of recursive constructions into sets.
Axiom schema: A rule that provides an axiom for each formula meeting specified conditions. Replacement is not one axiom but a family indexed by formulas.

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