Editing Axiom schema of specification (section)

== 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]].<ref name="GaborMath">{{Cite book |last=Toth |first=Gabor |url=https://books.google.com/books?id=bJhEEAAAQBAJ |title=Elements of Mathematics: A Problem-Centered Approach to History and Foundations |date=2021-09-23 |publisher=Springer Nature |isbn=978-3-030-75051-0 |pages=32 |language=en}}</ref>{{refn|group=lower-alpha|Suppes,<ref name="SuppesAxiomatic" /> cited earlier, derived it from the axiom schema of replacement alone (p. 237), but that's because he began his formulation of set theory by including the empty set as part of the definition of a set: his Definition 1, on page 19, states that <math>y \text{ is a set} \iff (\exists x) \ (x \in y \lor y = \emptyset)</math>.}}

The ''axiom schema of replacement'' says that, if a function <math>f</math> is definable by a formula <math>\varphi(x, y, p_1, \ldots, p_n)</math>, then for any set <math>A</math>, there exists a set <math>B = f(A) = \{ f(x) \mid x \in A \}</math>:

:<math>\begin{align}
&\forall x \, \forall y \, \forall z \, \forall p_1 \ldots \forall p_n [ \varphi(x, y, p_1, \ldots, p_n) \wedge \varphi(x, z, p_1, \ldots, p_n) \implies y = z ] \implies \\
&\forall A \, \exists B \, \forall y ( y \in B \iff \exists x ( x \in A \wedge \varphi(x, y, p_1, \ldots, p_n) ) )
\end{align}</math>.<ref name="GaborMath" />

To derive the axiom schema of specification, let <math>\varphi(x, p_1, \ldots, p_n)</math> be a formula and <math>z</math> a set, and define the function <math>f</math> such that <math>f(x) = x</math> if <math>\varphi(x, p_1, \ldots, p_n)</math> is true and <math>f(x) = u</math> if <math>\varphi(x, p_1, \ldots, p_n)</math> is false, where <math>u \in z</math> such that <math>\varphi(u, p_1, \ldots, p_n)</math> is true. Then the set <math>y</math> guaranteed by the axiom schema of replacement is precisely the set <math>y</math> required in the axiom schema of specification. If <math>u</math> does not exist, then <math>f(x)</math> in the axiom schema of specification is the empty set, whose existence (i.e., the axiom of empty set) is then needed.<ref name="GaborMath" />

For this reason, the axiom schema of specification is left out of some axiomatizations of '''ZF''' ([[Zermelo–Fraenkel set theory]]),<ref name=":3">{{Cite book |last=Bajnok |first=Béla |url=https://books.google.com/books?id=ZZUFEAAAQBAJ |title=An Invitation to Abstract Mathematics |date=2020-10-27 |publisher=Springer Nature |isbn=978-3-030-56174-1 |pages=138 |language=en}}</ref> although some authors, despite the redundancy, include both.<ref>{{Cite book |last=Vaught |first=Robert L. |url=https://books.google.com/books?id=sqxKHEwb5FkC |title=Set Theory: An Introduction |date=2001-08-28 |publisher=Springer Science & Business Media |isbn=978-0-8176-4256-3 |pages=67 |language=en}}</ref> Regardless, the axiom schema of specification is notable because it was in [[Ernst Zermelo|Zermelo]]'s original 1908 list of axioms, before [[Abraham Fraenkel|Fraenkel]] invented the axiom of replacement in 1922.<ref name=":3" /> Additionally, if one takes [[ZFC set theory|'''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).<ref>{{Cite book |last1=Kanovei |first1=Vladimir |url=https://books.google.com/books?id=GfDtCAAAQBAJ |title=Nonstandard Analysis, Axiomatically |last2=Reeken |first2=Michael |date=2013-03-09 |publisher=Springer Science & Business Media |isbn=978-3-662-08998-9 |pages=21 |language=en}}</ref>