In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, 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. Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Logic (from ancient Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Riverside Graphics Lab Open Directory Project: Computer Science Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: Computer science | Academic disciplines ... The word schema comes from the Greek word σχήμα (skhēma) that means shape or more generally plan. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

Suppose P is any predicate in one variable that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: In mathematics, a predicate is a relation. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...

or in words:

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

Note that there is one axiom for every such predicate P; thus, this is an axiom schema. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... SET may refer to: Secure electronic transaction, a protocol used for credit card processing, Simulated Emergency Test, an Amateur radio training exercise, Society for the Eradication of Television, Stock Exchange of Thailand, a national stock exchange of Thailand, SET Index, an index for Stock Exchange of Thailand This is a... In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ... In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ... AND Logic Gate Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ... In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...

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 {C ∈ A : P(C)}. Thus the essence of the axiom is: A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...

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

The axiom schema of specification is generally considered uncontroversial as far as it goes, and it or an equivalent appears in just about any alternative axiomatisation of set theory. Indeed, many alternative formulations of set theory try to find a way to use an even more generous axiom schema, while stopping short of the axiom schema of (unrestricted) comprehension mentioned below. In object-oriented programming, subclass is a class that is derived from another class or classes. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ...

Relation to the axiom schema of replacement

The axiom schema of separation can almost be derived from the axiom schema of replacement. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

First, recall this axiom schema:

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. In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, the empty set is the set with no elements. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ...

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 axiomatisations of set theory, as can be seen for example in the following sections.

Unrestricted comprehension

The axiom schema of unrestricted comprehension reads:

that is:

There exists a set B whose members are precisely those objects that satisfy the predicate P.

This set B is again unique, and is usually denoted as {C : P(C)}.

This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatisation was adopted. Unfortunately, it leads directly to Russell's paradox by taking P(C) to be (C is not in C). Therefore, no useful axiomatisation 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 serve as an additional replacement for the axiom schema of comprehension; each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy. Naive set theory1 is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ...

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 iff it belongs to some class E. In this theory, there is a theorem schema that reads: In foundations of mathematics, Von Neumann-Bernays-Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

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:

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 mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

In second order logic

In second-order logic, we can quantify over predicates, and 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 mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. ...

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; 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. In mathematical logic, the New Foundations (NF) of W. V. O. Quine is a candidate set theory, obtained from a streamlined version of the theory of types of Bertrand Russell. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...