Axiom schema of specification

For the separation axioms in topology, see separation axiom.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. It is also called the axiom schema of comprehension, although that term is also used for unrestricted comprehension, discussed below. Essentially, it says that any definable subclass of a set is a set.



One instance of the schema is included for each formula φ in the language of set theory with free variables among x, w1, ... , wn, A. So B is not free in φ. In the formal language of set theory, the axiom schema is:

\forall w_1,\ldots,w_n \, \forall A \, \exist B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \and \phi(x, w_1, \ldots, w_n, A) ] )

or in words:

Given any set A, there is a set B such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x.

Note that there is one axiom for every such predicate φ; 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 P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as {CA : P(C)}. Thus the essence of the axiom is:

Every subclass of a set that is defined by a predicate is itself a set.

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 axiom of comprehension of naive set theory. The Alternative Set Theory of Vopenka makes a specific point of allowing proper subclasses of sets, called semisets. 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 separation can almost be derived from the axiom schema of replacement.

First, recall this axiom schema:

\forall A \, \exist B \, \forall C \, ( C \in B \iff \exist D \, [ D \in A \and C = F(D) ] )

for any functional predicate F in one variable that doesn't use the symbols A, B, C or D. Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of A such that P(E) is true. Then the set B guaranteed by the axiom of replacement is precisely the set B required for the axiom of specification. The only problem is if no such E exists. But in this case, the set B required for the axiom of separation is the empty set, so the axiom of separation follows from the axiom of replacement together with the axiom of empty set.

For this reason, the axiom schema of separation is often left out of modern lists of the Zermelo-Fraenkel axioms. However, it's still important for historical considerations, and for comparison with alternative axiomatizations of set theory, as can be seen for example in the following sections.

Unrestricted comprehension

The axiom schema of comprehension (unrestricted) reads:

\forall w_1,\ldots,w_n \, \exist B \, \forall x \, ( x \in B \Leftrightarrow \phi(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_n)}.

This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatization was adopted. Unfortunately, it leads directly to Russell's paradox by taking φ(x) to be ¬(xx) (i.e., the property that set x is not a member of itself). Therefore, no useful axiomatization of set theory can use unrestricted comprehension, at least not with classical logic.

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 or the axiom of regularity) 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.

In NBG class 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:

\exist D: \forall C, C \isin D \harr \left(P\left(C\right) \and \exist E, C \isin E \right)

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.

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, \left(\exist E, A \isin E\right) \rarr \exist B, \left(\exist E, B \isin E \right) \and \forall C, C \isin B \harr \left(C \isin A \and C \isin D \right)

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.

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.


  • Paul Halmos, Naive set theory. Princeton, NJ: 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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Axiom schema of replacement — In set theory, the axiom schema of replacement is a schema of axioms in Zermelo Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite… …   Wikipedia

  • Axiom of limitation of size — In class theories, the axiom of limitation of size says that for any class C , C is a proper class (a class which is not a set (an element of other classes)) if and only if V (the class of all sets) can be mapped one to one into C .:forall C… …   Wikipedia

  • Axiom of infinity — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Fraenkel axioms,… …   Wikipedia

  • Axiom of union — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely… …   Wikipedia

  • Axiom — This article is about logical propositions. For other uses, see Axiom (disambiguation). In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self evident or to define and… …   Wikipedia

  • Schema — The word schema comes from the Greek word σχήμα (skhēma), which means shape, or more generally, plan . The Greek plural is σχήματα (skhēmata). In English, both schemas and schemata are used as plural forms, although the latter is the standard… …   Wikipedia

  • Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.ZFC consists of a single primitive ontological notion, that of… …   Wikipedia

  • General set theory — (GST) is George Boolos s (1998) name for a three axiom fragment of the canonical axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • List of axioms — This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self evidence. Individual axioms are almost always part of a larger axiomatic… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.