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Relation to the axiom schema of replacement

The axiom schema of specification is implied by the axiom schema of replacement together with the axiom of empty set.

The axiom schema of replacement says that, if a function f {\displaystyle f} is definable by a formula φ (x, y, p 1, …, p n) {\displaystyle \varphi (x,y,p_{1},\ldots,p_{n})}, then for any set A {\displaystyle A}, there exists a set B = f (A) = { f (x) ∣ x ∈ A } {\displaystyle B=f(A)=\{f(x)\mid x\in A\}}:

To derive the axiom schema of specification, let φ (x, p 1, …, p n) {\displaystyle \varphi (x,p_{1},\ldots,p_{n})} be a formula and z {\displaystyle z} a set, and define the function f {\displaystyle f} such that f (x) = x {\displaystyle f(x)=x} if φ (x, p 1, …, p n) {\displaystyle \varphi (x,p_{1},\ldots,p_{n})} is true and f (x) = u {\displaystyle f(x)=u} if φ (x, p 1, …, p n) {\displaystyle \varphi (x,p_{1},\ldots,p_{n})} is false, where u ∈ z {\displaystyle u\in z} such that φ (u, p 1, …, p n) {\displaystyle \varphi (u,p_{1},\ldots,p_{n})} is true. Then the set y {\displaystyle y} guaranteed by the axiom schema of replacement is precisely the set y {\displaystyle y} required in the axiom schema of specification. If u {\displaystyle u} does not exist, then f (x) {\displaystyle f(x)} in the axiom schema of specification is the empty set, whose existence (i.e., the axiom of empty set) is then needed.

For this reason, the axiom schema of specification is left out of some axiomatizations of ZF (Zermelo-Frankel) set theory, although some authors, despite the redundancy, include both. Regardless, the axiom schema of specification is notable because it was in Zermelo 's original 1908 list of axioms, before Fraenkel invented the axiom of replacement in 1922. Additionally, if one takes ZFC set theory (i.e., ZF with the axiom of choice), removes the axiom of replacement and the axiom of collection, but keeps the axiom schema of specification, one gets the weaker system of axioms called ZC (i.e., Zermelo's axioms, plus the axiom of choice).