Axiom schema of specification

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 ${\displaystyle f}$ is definable by a formula ${\displaystyle \phi (x,y,p_1,\dots ,p_n)}$, then for any set ${\displaystyle A}$, there exists a set ${\displaystyle B=f(A)=\{f(x)\mid x\in A\}}$:

${\displaystyle \begin{array}{cc}& \mathrm{\forall }x\mathrm{\forall }y\mathrm{\forall }z\mathrm{\forall }{p}_1\dots \mathrm{\forall }{p}_n[\phi (x,y,p_1,\dots ,p_n)\wedge \phi (x,z,p_1,\dots ,p_n)⟹y=z]⟹\\ & \mathrm{\forall }A\mathrm{\exists }B\mathrm{\forall }y(y\in B⟺\mathrm{\exists }x(x\in A\wedge \phi (x,y,p_1,\dots ,p_n)))\end{array}}$.

To derive the axiom schema of specification, let ${\displaystyle \phi (x,p_1,\dots ,p_n)}$ be a formula and ${\displaystyle z}$ a set, and define the function ${\displaystyle f}$ such that ${\displaystyle f(x)=x}$ if ${\displaystyle \phi (x,p_1,\dots ,p_n)}$ is true and ${\displaystyle f(x)=u}$ if ${\displaystyle \phi (x,p_1,\dots ,p_n)}$ is false, where ${\displaystyle u\in z}$ such that ${\displaystyle \phi (u,p_1,\dots ,p_n)}$ is true. Then the set ${\displaystyle y}$ guaranteed by the axiom schema of replacement is precisely the set ${\displaystyle y}$ required in the axiom schema of specification. If ${\displaystyle u}$ does not exist, then ${\displaystyle f(x)}$ 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–Fraenkel 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).

Unrestricted comprehension

The axiom schema of unrestricted comprehension reads:

$${\displaystyle \mathrm{\forall }{w}_1,\dots ,w_n\mathrm{\exists }B\mathrm{\forall }x(x\in B\iff \phi (x,w_1,\dots ,w_n))}$$

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, w₁, ..., w_b)}.

At an informal level, this axiom schema can be described as saying for any property or condition φ (the set's predicate), there exists a set ${\displaystyle \{x|\phi (x)\}}$ composed of all and only the objects which satisfy φ. For example, when φ is a tautology, the resulting set B is the universal set.

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 φ(x) to be ¬(x ∈ 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 from classical logic to intuitionistic logic does not help, as the proof of Russell's paradox is intuitionistically valid.

The axiom schema of specification can be viewed as a "restricted" version of this axiom schema, where φ can only be true for elements of another set A, thus forbidding the construction of sets that are "too large" such as the universal set. 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 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 or relative complement in positive set theory.

In NBG set theory

In von Neumann–Bernays–Gödel set theory, a distinction is made between sets and classes. A class C is a set if and only if it belongs to some class E. In this theory, there is a theorem schema that reads $${\displaystyle \mathrm{\exists }D\mathrm{\forall }C([C\in D]⟺[P(C)\wedge \mathrm{\exists }E(C\in E)]),}$$

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 $${\displaystyle \mathrm{\forall }D\mathrm{\forall }A(\mathrm{\exists }E[A\in E]⟹\mathrm{\exists }B[\mathrm{\exists }E(B\in E)\wedge \mathrm{\forall }C(C\in B⟺[C\in A\wedge C\in D])]),}$$

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

The intersection of 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 $${\displaystyle \mathrm{\forall }A\mathrm{\forall }B([\mathrm{\exists }E(A\in E)\wedge \mathrm{\forall }C(C\in B⟹C\in A)]⟹\mathrm{\exists }E[B\in E]),}$$

that is,

A subclass of a set is a set.

In higher-order settings

In a 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

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 (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, see stratification.