In
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 mathematics, ...
and the branches of
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
that use it, the axiom of extensionality, or axiom of extension, is one of the
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of
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 such as ...
. It says that sets having the same elements are the same set.
Formal statement
In the
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
of the Zermelo–Fraenkel axioms, the axiom reads:
:
or in words:
:
Given any
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
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'' and any set ''B'', if for every set ''X'', ''X'' is a member of ''A''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
''X'' is a member of ''B'', then ''A'' is
equal to ''B''.
:(It is not really essential that ''X'' here be a ''set'' — but in ZF, everything is. See
Ur-elements below for when this is violated.)
The converse,
of this axiom follows from the substitution property of
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
.
Interpretation
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that ''A'' and ''B'' have precisely the same members.
Thus, what the axiom is really saying is that two sets are equal
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
they have precisely the same members.
The essence of this is:
:A set is determined uniquely by its members.
The axiom of extensionality can be used with any statement of the form
,
where ''P'' is any unary
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
that does not mention ''A'', to define a unique set
whose members are precisely the sets satisfying the predicate
.
We can then introduce a new symbol for
; it's in this way that
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
s in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.
The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.
However, it may require modifications for some purposes, as below.
In predicate logic without equality
The axiom given above assumes that equality is a primitive symbol in
predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
.
Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a ''definition'' of equality.
Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is the substitution property,
:
and it becomes ''this'' axiom that is referred to as the axiom of extensionality in this context.
In set theory with ur-elements
An
ur-element
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual.
Theory
There ...
is a member of a set that is not itself a set.
In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory.
Ur-elements can be treated as a different
logical type
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foun ...
from sets; in this case,
makes no sense if
is an ur-element, so the axiom of extensionality simply applies only to sets.
Alternatively, in untyped logic, we can require
to be false whenever
is an ur-element.
In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
.
To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:
:
That is:
:Given any set ''A'' and any set ''B'', ''if ''A'' is a nonempty set'' (that is, if there exists a member ''X'' of ''A''), ''then'' if ''A'' and ''B'' have precisely the same members, then they are equal.
Yet another alternative in untyped logic is to define
itself to be the only element of
whenever
is an ur-element. While this approach can serve to preserve the axiom of extensionality, 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 non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the axi ...
will need an adjustment instead.
See also
*
Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
for a general overview.
References
*
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
, ''Naive set theory''. Princeton, NJ: 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. .
{{Set theory
Axioms of set theory
Urelements