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{{short description|Concept in axiomatic set theory}}
{{redirect|Axiom of separation|the separation axioms in topology|separation axiom}}

In many popular versions of [[axiomatic set theory]], the '''axiom schema of specification''',<ref name=":1">{{Cite web |title=AxiomaticSetTheory |url=https://www.cs.yale.edu/homes/aspnes/pinewiki/AxiomaticSetTheory.html |access-date=2024-06-08 |website=www.cs.yale.edu |at=Axiom Schema of Specification}}</ref> also known as the '''axiom schema of separation''' (''Aussonderungsaxiom''),<ref name="SuppesAxiomatic">{{Cite book |last=Suppes |first=Patrick |url=https://books.google.com/books?id=sxr4LrgJGeAC |title=Axiomatic Set Theory |date=1972-01-01 |publisher=Courier Corporation |isbn=978-0-486-61630-8 |pages=6,19,21,237 |language=en |quote=}}</ref> '''subset axiom<ref name=":0">{{Cite book |last=Cunningham |first=Daniel W. |title=Set theory: a first course |date=2016 |publisher=Cambridge University Press |isbn=978-1-107-12032-7 |series=Cambridge mathematical textbooks |location=New York, NY |pages=22,24-25,29}}</ref>''', '''axiom of class construction''',<ref>{{Cite book |last=Pinter |first=Charles C. |url=https://books.google.com/books?id=iUT_AwAAQBAJ |title=A Book of Set Theory |date=2014-06-01 |publisher=Courier Corporation |isbn=978-0-486-79549-2 |pages=27 |language=en}}</ref> or '''axiom schema of restricted comprehension''' is an [[axiom schema]]. Essentially, it says that any definable [[subclass (set theory)|subclass]] of a set is a set.

Some mathematicians call it the '''axiom schema of comprehension''', although others use that term for '''''unrestricted'' comprehension''', discussed below.

Because restricting comprehension avoided [[Russell's paradox]], several mathematicians including [[Ernst Zermelo|Zermelo]], [[Abraham Fraenkel|Fraenkel]], and [[Gödel]] considered it the most important axiom of set theory.<ref name="Ebbinghaus2007">{{cite book|author=Heinz-Dieter Ebbinghaus|title=Ernst Zermelo: An Approach to His Life and Work|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-49553-6|page=88}}</ref>

== Statement ==
One instance of the schema is included for each [[Well-formed formula|formula]] <math>\varphi</math> in the language of set theory with [[free variables]] among <math>x,w_1,w_2,\ldots,w_n,A</math> . So <math>B</math> does not occur free in <math>\varphi</math>. In the formal language of set theory, the axiom schema is:
:<math>\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \land \varphi(x, w_1, \ldots, w_n , A) ] )</math>

or in words:
: Given any [[Set (mathematics)|set]] ''A'', [[Existential quantification|there is]] a set ''B'' (a subset of ''A'') such that, given any set ''x'', ''x'' is a member of ''B'' [[if and only if]] ''x'' is a member of ''A'' [[logical conjunction|and]] <math>\varphi</math> holds for ''x''.
Note that there is one axiom for every such [[predicate (mathematics)|predicate]] <math>\varphi</math>; thus, this is an [[axiom schema]].

To understand this axiom schema, note that the set ''B'' must be a [[subset]] of ''A''. Thus, what the axiom schema is really saying is that, given a set ''A'' and a predicate <math>\varphi</math>, we can find a subset ''B'' of ''A'' whose members are precisely the members of ''A'' that satisfy <math>\varphi</math>. By the [[axiom of extensionality]] this set is unique. We usually denote this set using [[set-builder notation]] as <math>B = \{x\in A | \varphi(x) \}</math>. Thus the essence of the axiom is:
: Every [[Subclass (set theory)|subclass]] of a set that is defined by a predicate is itself a set.

The preceding form of separation was introduced in 1930 by [[Thoralf Skolem]] as a refinement of a previous, non-first-order<ref>F. R. Drake, ''Set Theory: An Introduction to Large Cardinals'' (1974), pp.12--13. ISBN 0 444 10535 2.</ref> form by Zermelo.<ref>W. V. O. Quine, ''Mathematical Logic'' (1981), p.164. Harvard University Press, 0-674-55451-5</ref> The axiom schema of specification is characteristic of systems of [[axiomatic set theory]] related to the usual set theory [[ZFC]], but does not usually appear in radically different systems of [[alternative set theory]]. For example, [[New Foundations]] and [[positive set theory]] use different restrictions of the [[#Unrestricted comprehension|axiom of comprehension]] of [[naive set theory]]. The [[Alternative Set Theory]] of Vopenka makes a specific point of allowing proper subclasses of sets, called [[semiset]]s. Even in systems related to ZFC, this scheme is sometimes restricted to formulas with bounded quantifiers, as in [[Kripke–Platek set theory with urelements]].

== 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>

== Unrestricted comprehension ==
{{Unreferenced section|date=June 2024}}
{{also|Basic Law V}}
The '''axiom schema of unrestricted comprehension''' reads:

<math display="block">\forall w_1,\ldots,w_n \, \exists B \, \forall x \, ( x \in B \Leftrightarrow \varphi(x, w_1, \ldots, w_n) )</math>

that is:
{{block indent|There exists a set {{mvar|B}} whose members are precisely those objects that satisfy the predicate {{mvar|φ}}.}}

This set {{mvar|B}} is again unique, and is usually denoted as {{math|{{{var|x}} : {{var|φ}}({{var|x}}, {{mvar|w}}{{sub|1}}, ..., {{var|w}}{{sub|{{mvar|b}}}})}.}}

In an unsorted material set theory, the axiom or rule of '''full''' or '''unrestricted comprehension''' says that for any property {{mvar|P}}, there exists a set <math>\{x | P(x)\}</math> of all objects satisfying {{mvar|P}}.<ref>{{Cite web |title=axiom of full comprehension in nLab |url=https://ncatlab.org/nlab/show/axiom+of+full+comprehension?t&utm_source=perplexity#definition |access-date=2024-11-07 |website=ncatlab.org |language=en}}</ref>

This axiom schema was tacitly used in the early days of [[naive set theory]], before a strict axiomatization was adopted. However, it was later discovered to lead directly to [[Russell's paradox]], by taking {{math|{{var|φ}}({{var|x}})}} to be {{math|¬({{var|x}} ∈ {{var|x}})}} (i.e., the property that set {{mvar|x}} is not a member of itself). Therefore, no useful [[axiomatization]] of set theory can use unrestricted comprehension. Passing from [[classical logic]] to [[intuitionistic logic]] does not help, as the proof of Russell's paradox is intuitionistically valid.

Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo–Fraenkel axioms (but not the [[axiom of extensionality]], the [[axiom of regularity]], or the [[axiom of choice]]) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification—each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i.e. it is a special case of the axiom schema of comprehension.

It is also possible to prevent the schema from being inconsistent by restricting which formulae it can be applied to, such as only [[Stratification (mathematics)|stratified]] formulae in [[New Foundations]] (see below) or only positive formulae (formulae with only conjunction, disjunction, quantification and atomic formulae) in [[positive set theory]]. Positive formulae, however, typically are unable to express certain things that most theories can; for instance, there is no [[Complement (set theory)|complement]] or [[relative complement]] in positive set theory.

== In NBG set theory ==
{{Unreferenced section|date=June 2024}}
In [[von Neumann–Bernays–Gödel set theory]], a distinction is made between sets and [[class (set theory)|classes]]. A class {{mvar|C}} is a set if and only if it belongs to some class {{mvar|E}}. In this theory, there is a [[theorem]] schema that reads
<math display="block">\exists D \forall C \, ( [ C \in D ] \iff [ P (C) \land \exists E \, ( C \in E ) ] ) \,,</math>

that is,
{{block indent|There is a class {{mvar|D}} such that any class {{mvar|C}} is a member of {{mvar|D}} if and only if {{mvar|C}} is a set that satisfies {{mvar|P}}.}}

provided that the quantifiers in the predicate {{mvar|P}} are restricted to sets.

This theorem schema is itself a restricted form of comprehension, which avoids Russell's paradox because of the requirement that {{mvar|C}} be a set. Then specification for sets themselves can be written as a single axiom
<math display="block">\forall D \forall A \, ( \exists E \, [ A \in E ] \implies \exists B \, [ \exists E \, ( B \in E ) \land \forall C \, ( C \in B \iff [ C \in A \land C \in D ] ) ] ) \,,</math>

that is,
{{block indent|Given any class {{mvar|D}} and any set {{mvar|A}}, there is a set {{mvar|B}} whose members are precisely those classes that are members of both {{mvar|A}} and {{mvar|D}}.}}

or even more simply
{{block indent|The [[intersection (set theory)|intersection]] of a class {{mvar|D}} and a set {{mvar|A}} is itself a set {{mvar|B}}.}}

In this axiom, the predicate {{mvar|P}} is replaced by the class {{mvar|D}}, which can be quantified over. Another simpler axiom which achieves the same effect is
<math display="block">\forall A \forall B \, ( [ \exists E \, ( A \in E ) \land \forall C \, ( C \in B \implies C \in A ) ] \implies \exists E \, [ B \in E ] ) \,,</math>

that is,
{{block indent|A subclass of a set is a set.}}

== In higher-order settings ==
{{Unreferenced section|date=June 2024}}
In a [[type theory|typed]] language where we can quantify over predicates, the axiom schema of specification becomes a simple axiom. This is much the same trick as was used in the NBG axioms of the previous section, where the predicate was replaced by a class that was then quantified over.

In [[second-order logic]] and [[higher-order logic]] with higher-order semantics, the axiom of specification is a logical validity and does not need to be explicitly included in a theory.

== In Quine's New Foundations ==
{{Unreferenced section|date=June 2024}}
In the [[New Foundations]] approach to set theory pioneered by [[W. V. O. Quine]], the axiom of comprehension for a given predicate takes the unrestricted form, but the predicates that may be used in the schema are themselves restricted. The predicate ({{mvar|C}} is not in {{mvar|C}}) is forbidden, because the same symbol {{mvar|C}} appears on both sides of the membership symbol (and so at different "relative types"); thus, Russell's paradox is avoided. However, by taking {{math|{{var|P}}({{var|C}})}} to be {{math|1=({{var|C}} = {{var|C}})}}, which is allowed, we can form a set of all sets. For details, see [[stratification (mathematics)|stratification]].

==References==
{{reflist}}
==Further reading==
{{refbegin}}
* {{cite book | last1=Crossley | first1=J.bN. | last2=Ash | first2=C. J. | last3=Brickhill | first3=C. J. | last4=Stillwell | first4=J. C. | last5=Williams | first5=N. H. | title=What is mathematical logic? | zbl=0251.02001 | location=London-Oxford-New York | publisher=[[Oxford University Press]] | year=1972 | isbn=0-19-888087-1 }}
*[[Paul Halmos|Halmos, Paul]], ''[[Naive Set Theory (book)|Naive Set Theory]]''. Princeton, New Jersey: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. {{ISBN|0-387-90092-6}} (Springer-Verlag edition).
*Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. {{ISBN|3-540-44085-2}}.
*Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. {{ISBN|0-444-86839-9}}.
{{refend}}
==Notes==
{{reflist|group=lower-alpha}}

{{Set theory}}

[[Category:Axioms of set theory]]

Axiom schema of specification - Revision history

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