Axiom schema of specification

In many popular versions ofaxiomatic set theory,theaxiom schema of specification,^[1]also known as theaxiom schema of separation(Aussonderung Axiom),^[2]subset axiom^[3]oraxiom schema of restricted comprehensionis anaxiom schema.Essentially, it says that any definablesubclassof a set is a set.

Some mathematicians call it theaxiom schema of comprehension,although others use that term forunrestrictedcomprehension,discussed below.

Because restricting comprehension avoidedRussell's paradox,several mathematicians includingZermelo,Fraenkel,andGödelconsidered it the most important axiom of set theory.^[4]

Statement[edit]

One instance of the schema is included for eachformula $\varphi (x)$ in the language of set theory with $x$ as a free variable. So $S$ does not occur free in $\varphi (x)$ .^[3]^[2]^[5]In the formal language of set theory, the axiom schema is:

\forall A\,\exists S\,\forall x\,{\big (}x\in S\iff (x\in A\wedge \varphi (x)){\big )}

^[3]^[1]^[5]

or in words:

Let

\varphi (x)

be a formula. For every set

A

there exists a set

S

that consists of all the elements

x\in A

such that

\varphi (x)

holds.^[3]

Note that there is one axiom for every suchpredicate $\varphi (x)$ ;thus, this is anaxiom schema.^[3]^[1]

To understand this axiom schema, note that the set $S$ must be asubsetofA.Thus, what the axiom schema is really saying is that, given a set $A$ and a predicate $\varphi (x)$ ,we can find a subset $S$ ofAwhose members are precisely the members ofAthat satisfy $\varphi (x)$ .By theaxiom of extensionalitythis set is unique. We usually denote this set usingset-builder notationas $S=\{x\in A|\varphi (x)\}$ .Thus the essence of the axiom is:

Everysubclassof a set that is defined by a predicate is itself a set.

The preceding form of separation was introduced in 1930 byThoralf Skolemas a refinement of a previous, non-first-order^[6]form by Zermelo.^[7]The axiom schema of specification is characteristic of systems ofaxiomatic set theoryrelated to the usual set theoryZFC,but does not usually appear in radically different systems ofalternative set theory.For example,New Foundationsandpositive set theoryuse different restrictions of theaxiom of comprehensionofnaive set theory.TheAlternative Set Theoryof Vopenka makes a specific point of allowing proper subclasses of sets, calledsemisets.Even in systems related to ZFC, this scheme is sometimes restricted to formulas with bounded quantifiers, as inKripke–Platek set theory with urelements.

Relation to the axiom schema of replacement[edit]

The axiom schema of specification is implied by theaxiom schema of replacementtogether with theaxiom of empty set.^[8]^[a]

Theaxiom schema of replacementsays that, if a function $f$ is definable by a formula $\varphi (x,y,p_{1},\ldots,p_{n})$ ,then for any set $A$ ,there exists a set $B=f(A)=\{f(x)\mid x\in A\}$ :

{\begin{aligned}&\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{aligned}}

.^[8]

To derive the axiom schema of specification, let $\varphi (x,p_{1},\ldots,p_{n})$ be a formula and $z$ a set, and define the function $f$ such that $f(x)=x$ if $\varphi (x,p_{1},\ldots,p_{n})$ is true and $f(x)=u$ if $\varphi (x,p_{1},\ldots,p_{n})$ is false, where $u\in z$ such that $\varphi (u,p_{1},\ldots,p_{n})$ is true. Then the set $y$ guaranteed by the axiom schema of replacement is precisely the set $y$ required in the axiom schema of specification. If $u$ does not exist, then $f(x)$ in the axiom schema of specification is the empty set, whose existence (i.e., the axiom of empty set) is then needed.^[8]

For this reason, the axiom schema of specification is left out of some axiomatizations ofZF (Zermelo-Frankel) set theory,^[9]although some authors, despite the redundancy, include both.^[10]Regardless, the axiom schema of specification is notable because it was inZermelo's original 1908 list of axioms, beforeFraenkelinvented the axiom of replacement in 1922.^[9]Additionally, if one takesZFC set theory(i.e., ZF with the axiom of choice), removes the axiom of replacement and theaxiom of collection,but keeps the axiom schema of specification, one gets the weaker system of axioms calledZC(i.e., Zermelo's axioms, plus the axiom of choice).^[11]

Unrestricted comprehension[edit]

Theaxiom schema of unrestricted comprehensionreads:

$\forall w_{1},\ldots,w_{n}\,\exists B\,\forall x\,(x\in B\Leftrightarrow \varphi (x,w_{1},\ldots,w_{n}))$

that is:

There exists a set

B

whose members are precisely those objects that satisfy the predicate

φ

.

This set $B$ is again unique, and is usually denoted as ${x : φ (x, w 1,..., w b)}.$

This axiom schema was tacitly used in the early days ofnaive set theory,before a strict axiomatization was adopted. However, it was later discovered to lead directly toRussell's paradox,by taking $φ (x)$ to be $\neg(x \in x)$ (i.e., the property that set $x$ is not a member of itself). Therefore, no useful axiomatization of set theory can use unrestricted comprehension. Passing fromclassical logictointuitionistic logicdoes 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 theaxiom of extensionality,theaxiom of regularity,or theaxiom 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 onlystratifiedformulae inNew Foundations(see below) or only positive formulae (formulae with only conjunction, disjunction, quantification and atomic formulae) inpositive set theory.Positive formulae, however, typically are unable to express certain things that most theories can; for instance, there is nocomplementor relative complement in positive set theory.

In NBG class theory[edit]

Invon Neumann–Bernays–Gödel set theory,a distinction is made between sets andclasses.A class $C$ is a set if and only if it belongs to some class $E$ .In this theory, there is atheoremschema that reads $\exists D\forall C\,([C\in D]\iff [P(C)\land \exists E\,(C\in E)])\,,$

that is,

There is a class

D

such that any class

C

is a member of

D

if and only if

C

is a set that satisfies

P

.

provided that the quantifiers in the predicate $P$ are restricted to sets.

This theorem schema is itself a restricted form of comprehension, which avoids Russell's paradox because of the requirement that $C$ be a set. Then specification for sets themselves can be written as a single axiom $\forall D\forall A\,(\exists E\,[A\in E]\implies \exists B\,[\exists E\,(B\in E)\land \forall C\,(C\in B\iff [C\in A\land C\in D])])\,,$

that is,

Given any class

D

and any set

A

,there is a set

B

whose members are precisely those classes that are members of both

A

and

D

.

or even more simply

Theintersectionof a class

D

and a set

A

is itself a set

B

.

In this axiom, the predicate $P$ is replaced by the class $D$ ,which can be quantified over. Another simpler axiom which achieves the same effect is $\forall A\forall B\,([\exists E\,(A\in E)\land \forall C\,(C\in B\implies C\in A)]\implies \exists E\,[B\in E])\,,$

that is,

A subclass of a set is a set.

In higher-order settings[edit]

In atypedlanguage 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.

Insecond-order logicandhigher-order logicwith 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[edit]

In theNew Foundationsapproach to set theory pioneered byW. 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 ( $C$ is not in $C$ ) is forbidden, because the same symbol $C$ appears on both sides of the membership symbol (and so at different "relative types" ); thus, Russell's paradox is avoided. However, by taking $P (C)$ to be $(C = C)$ ,which is allowed, we can form a set of all sets. For details, seestratification.

References[edit]

^^a ^b ^c"AxiomaticSetTheory".cs.yale.edu.Axiom Schema of Specification.Retrieved2024-06-08.
^^a ^b ^cSuppes, Patrick (1972-01-01).Axiomatic Set Theory.Courier Corporation. pp. 6, 19, 21, 237.ISBN 978-0-486-61630-8.
^^a ^b ^c ^d ^eCunningham, Daniel W. (2016).Set theory: a first course.Cambridge mathematical textbooks. New York, NY: Cambridge University Press. pp. 22, 24–25, 29.ISBN 978-1-107-12032-7.
^Heinz-Dieter Ebbinghaus (2007).Ernst Zermelo: An Approach to His Life and Work.Springer Science & Business Media. p. 88.ISBN 978-3-540-49553-6.
^^a ^bDeVidi, David; Hallett, Michael; Clark, Peter (2011-03-23).Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell.Springer Science & Business Media. p. 206.ISBN 978-94-007-0214-1.
^F. R. Drake,Set Theory: An Introduction to Large Cardinals (1974), pp.12--13. ISBN 0 444 10535 2.
^W. V. O. Quine,Mathematical Logic(1981), p.164. Harvard University Press, 0-674-55451-5
^^a ^b ^cToth, Gabor (2021-09-23).Elements of Mathematics: A Problem-Centered Approach to History and Foundations.Springer Nature. p. 32.ISBN 978-3-030-75051-0.
^^a ^bBajnok, Béla (2020-10-27).An Invitation to Abstract Mathematics.Springer Nature. p. 138.ISBN 978-3-030-56174-1.
^Vaught, Robert L. (2001-08-28).Set Theory: An Introduction.Springer Science & Business Media. p. 67.ISBN 978-0-8176-4256-3.
^Kanovei, Vladimir; Reeken, Michael (2013-03-09).Nonstandard Analysis, Axiomatically.Springer Science & Business Media. p. 21.ISBN 978-3-662-08998-9.

Notes[edit]

^Suppes,^[2]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 $y{\text{ is a set}}\iff (\exists x)\ (x\in y\lor y=\emptyset )$ .

[:1-1] "AxiomaticSetTheory".cs.yale.edu.Axiom Schema of Specification.Retrieved2024-06-08.

[SuppesAxiomatic-2] Suppes, Patrick (1972-01-01).Axiomatic Set Theory.Courier Corporation. pp. 6, 19, 21, 237.ISBN 978-0-486-61630-8.

[:0-3] Cunningham, Daniel W. (2016).Set theory: a first course.Cambridge mathematical textbooks. New York, NY: Cambridge University Press. pp. 22, 24–25, 29.ISBN 978-1-107-12032-7.

[Ebbinghaus2007-4] Heinz-Dieter Ebbinghaus (2007).Ernst Zermelo: An Approach to His Life and Work.Springer Science & Business Media. p. 88.ISBN 978-3-540-49553-6.

[:2-5] DeVidi, David; Hallett, Michael; Clark, Peter (2011-03-23).Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell.Springer Science & Business Media. p. 206.ISBN 978-94-007-0214-1.

[6] F. R. Drake,Set Theory: An Introduction to Large Cardinals (1974), pp.12--13. ISBN 0 444 10535 2.

[7] W. V. O. Quine,Mathematical Logic(1981), p.164. Harvard University Press, 0-674-55451-5

[GaborMath-8] Toth, Gabor (2021-09-23).Elements of Mathematics: A Problem-Centered Approach to History and Foundations.Springer Nature. p. 32.ISBN 978-3-030-75051-0.

[:3-10] Bajnok, Béla (2020-10-27).An Invitation to Abstract Mathematics.Springer Nature. p. 138.ISBN 978-3-030-56174-1.

[11] Vaught, Robert L. (2001-08-28).Set Theory: An Introduction.Springer Science & Business Media. p. 67.ISBN 978-0-8176-4256-3.

[12] Kanovei, Vladimir; Reeken, Michael (2013-03-09).Nonstandard Analysis, Axiomatically.Springer Science & Business Media. p. 21.ISBN 978-3-662-08998-9.

[9] Suppes,^[2]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 $y{\text{ is a set}}\iff (\exists x)\ (x\in y\lor y=\emptyset )$ .

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v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement(i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number(large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Settypes	Amorphous Countable Empty Finite(hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite(Dedekind-infinite) Recursive Singleton Subset·Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

Statement[edit]

Relation to the axiom schema of replacement[edit]

Unrestricted comprehension[edit]

In NBG class theory[edit]

In higher-order settings[edit]

In Quine's New Foundations[edit]

References[edit]

Further reading[edit]

Notes[edit]