Axiom of extensionality
axiomatic set theoryand the branches of logic, mathematics, and computer sciencethat use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.
formal languageof the Zermelo-Fraenkel axioms, the axiom reads:
or in words::
Given anyset "A" and any set "B", if for every set "C", "C" is a member of "A" if and only if"C" is a member of "B", then "A" is equal to "B".(It is not really essential that "C" 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.
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 ifthey 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 that does not mention "A" or "B", 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
definitions in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.
The axiom of extensionality is generally considered uncontroversial, 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.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:and it becomes "this" axiom that is referred to as the axiom of extensionality in this context.
In set theory with ur-elements
ur-elementis a member of a set that is not itself a set.In the Zermelo-Fraenkel axioms, there are no ur-elements, but some alternative axiomatisations of set theory have them.Ur-elements can be treated as a different logical typefrom 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 imply that every ur-element is equal to the
empty set.To avoid this, 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 "C" 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 regularitywill need an adjustment instead.
*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.
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