In set theory, the axiom schema of replacement is a schema of axioms in ZermeloFraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZFC.
The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a bijection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable bijections, which are identified with their defining formulas.
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Statement
Suppose P is a definable binary relation (which may be a proper class) such that for every set x there is a unique set y such that P(x,y) holds. There is a corresponding definable function F_{P}, where F_{P}(X) = Y if and only if P(X,Y); F will also be a proper class if P is. Consider the (possibly proper) class B defined such for every set y, y is in B if and only if there is an x in A with F_{P}(x) = y. B is called the image of A under F_{P}, and denoted F_{P}(A) or (using setbuilder notation) {F_{P}(x) : x ∈ A}.
The axiom schema of replacement states that if F is a definable class function, as above, and A is any set, then the image F(A) is also a set. This can be seen as a principle of smallness: the axiom states that if A is small enough to be a set, then F(A) is also small enough to be a set. It is implied by the stronger axiom of limitation of size.
Because it is impossible to quantify over definable functions in firstorder logic, one instance of the schema is included for each formula φ in the language of set theory with free variables among w_{1}, ... , w_{n}, A, x, y; but B is not free in φ. In the formal language of set theory, the axiom schema is:
Axiom schema of collection
The axiom schema of collection is closely related to and frequently confused with the axiom schema of replacement. While replacement says that the image itself is a set, collection merely says that a superclass of the image is a set. In other words, the resulting set, B, is not required to be minimal.
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