axiom schema of specification
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In many popular versions of
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, the axiom schema of specification, also known as the axiom schema of separation (''Aussonderungsaxiom''), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variabl ...
. Essentially, it says that any definable 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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains ...
, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.


Statement

One instance of the schema is included for each
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
\varphi in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in \varphi. In the formal language of set theory, the axiom schema is: :\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) ) or in words: : Given any
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''A'', 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''x'' is a member of ''A'' and \varphi holds for ''x''. Note that there is one axiom for every such predicate \varphi; thus, this is an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variabl ...
. To understand this axiom schema, note that the set ''B'' must be a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of ''A''. Thus, what the axiom schema is really saying is that, given a set ''A'' and a predicate \varphi, we can find a subset ''B'' of ''A'' whose members are precisely the members of ''A'' that satisfy \varphi. By the
axiom of extensionality The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the ...
this set is unique. We usually denote this set using set-builder notation as B = \. Thus the essence of the axiom is: : Every 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 Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skole ...
as a refinement of a previous, non-first-order form by Zermelo. The axiom schema of specification is characteristic of systems of
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
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 specification is implied by the axiom schema of replacement together with the
axiom of empty set In axiomatic set theory, the axiom of empty set, also called the axiom of null set and the axiom of existence, is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of g ...
. The ''axiom schema of replacement'' says 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) = \: :\begin &\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. 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. For this reason, the axiom schema of specification is left out of some axiomatizations of ZF (
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
), 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: \forall w_1,\ldots,w_n \, \exists B \, \forall x \, ( x \in B \Leftrightarrow \varphi(x, w_1, \ldots, w_n) ) that is: This set is again unique, and is usually denoted as In an unsorted material set theory, the axiom or rule of full or unrestricted comprehension says that for any property ''P'', there exists a set of all objects satisfying ''P.'' 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 In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains ...
, by taking to be (i.e., the property that set is not a member of itself). Therefore, no useful axiomatization of set theory can use unrestricted comprehension. Passing from
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
to
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
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 extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the ...
, the
axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoin ...
, or the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
) 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 class theory

In von Neumann–Bernays–Gödel set theory, a distinction is made between sets and classes. A class is a set if and only if it belongs to some class . In this theory, there is a
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
schema that reads \exists D \forall C \, ( C \in D \iff P (C) \land \exists E \, ( C \in E ) ) \,, that is, provided that the quantifiers in the predicate are restricted to sets. This theorem schema is itself a restricted form of comprehension, which avoids Russell's paradox because of the requirement that 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, or even more simply In this axiom, the predicate is replaced by the class , 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,


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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
and
higher-order logic In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are m ...
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 ( is not in ) is forbidden, because the same symbol appears on both sides of the membership symbol (and so at different "relative types"); thus, Russell's paradox is avoided. However, by taking to be , which is allowed, we can form a set of all sets. For details, see stratification.


References


Further reading

* * Halmos, Paul, '' Naive Set Theory''. Princeton, New Jersey: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . *Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. .


Notes

{{Set theory Axioms of set theory