Theaxiom of extensionality,^[1][2] also called theaxiom of extent,^[3][4] is anaxiom used in many forms ofaxiomatic set theory, such asZermelo–Fraenkel set theory.^[5][6] The axiom defines what aset is.^[1] Informally, the axiom means that the twosetsA andB are equalif and only ifA andB have the same members.

The termextensionality, as used in'Axiom of Extensionality' has its roots in logic. Anintensional definition describes thenecessary and sufficient conditions for a term to apply to an object. For example: "Aneven number is aninteger which isdivisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, theextension of apredicate is the set of all things for which the predicate is true.^[7]

The logical term was introduced to set theory in 1893,Gottlob Frege attempted to use this idea of an extension formally in hisBasic Laws of Arithmetic (German:Grundgesetze der Arithmetik),^[8][9] where, if${\displaystyle F}$ is a predicate, its extension (German:Umfang)${\displaystyle \epsilon F}$, is the set of all objects satisfying${\displaystyle F}$.^[10] For example if${\displaystyle F(x)}$ is "x is even" then${\displaystyle \epsilon F}$ is the set${\displaystyle \{\cdots ,-4,-2,0,2,4,\cdots \}}$. In his work, he defined his infamousBasic Law V as:^[11]$${\displaystyle \epsilon F=\epsilon G\equiv \mathrm{\forall }x(F(x)\equiv G(x))}$$Stating that if two predicates have the same extensions (they are satisfied by the same set of objects) then they are logically equivalent, however, it was determined later that this axiom led toRussell's paradox. The first explicit statement of the modern Axiom of Extensionality was in 1908 by Ernst Zermelo in a paper on thewell-ordering theorem, where he presented the first axiomatic set theory, now calledZermelo set theory, which became the basis of modern set theories.^[12] The specific term for "Extensionality" used by Zermelo was "Bestimmtheit".The specific English term "extensionality" only became common in mathematical and logical texts in the 1920s and 1930s,^[13] particularly with the formalization of logic and set theory by figures likeAlfred Tarski andJohn von Neumann.

In theformal language of the Zermelo–Fraenkel axioms, the axiom reads:

${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[\mathrm{\forall }z(z\in x\leftrightarrow z\in y)\to x=y]}$^[14][15][16]

or in words:

If the sets${\displaystyle x}$ and${\displaystyle y}$ have the same members, then they are the same set.^[14][1]

Inpure set theory, all members of sets are themselves sets, but not in set theory withurelements. The axiom's usefulness can be seen from the fact that, if one accepts that${\displaystyle \mathrm{\exists }A\mathrm{\forall }x(x\in A⟺\mathrm{\Phi }(x))}$, where${\displaystyle A}$ is a set and${\displaystyle \mathrm{\Phi }(x)}$ is a formula that${\displaystyle x}$occurs free in but${\displaystyle A}$ doesn't, then the axiom assures that there is a unique set${\displaystyle A}$ whose members are precisely whatever objects (urelements or sets, as the case may be) satisfy the formula${\displaystyle \mathrm{\Phi }(x)}$.

The converse of the axiom,${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[x=y\to \mathrm{\forall }z(z\in x\leftrightarrow z\in y)]}$, follows from thesubstitution property ofequality. Despite this, the axiom is sometimes given directly as abiconditional, i.e., as${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[\mathrm{\forall }z(z\in x\leftrightarrow z\in y)\leftrightarrow x=y]}$.^[1]

Quine'sNew Foundations (NF) set theory, in Quine's original presentations of it, treats the symbol${\displaystyle =}$ for equality or identity as shorthand either for "if a set contains the left side of the equals sign as a member, then it also contains the right side of the equals sign as a member" (as defined in 1937), or for "an object is an element of the set on the left side of the equals sign if, and only if, it is also an element of the set on the right side of the equals sign" (as defined in 1951). That is,${\displaystyle x=y}$ is treated as shorthand either for${\displaystyle \mathrm{\forall }z(x\in z\to y\in z)}$, as in the original 1937 paper, or for${\displaystyle \mathrm{\forall }z(z\in x\leftrightarrow z\in y)}$, as in Quine'sMathematical Logic (1951). The second version of the definition is exactly equivalent to theantecedent of the ZF axiom of extensionality, and the first version of the definition is still very similar to it. By contrast, however, the ZF set theory takes the symbol${\displaystyle =}$ for identity or equality as a primitive symbol of the formal language, and defines the axiom of extensionality in terms of it. (In this paragraph, the statements of both versions of the definition were paraphrases, and quotation marks were only used to set the statements apart.)

In Quine'sNew Foundations for Mathematical Logic (1937), the original paper of NF, the name "principle of extensionality" is given to the postulate P1,${\displaystyle ((x\subset y)\supset ((y\subset x)\supset (x=y)))}$,^[17] which, for readability, may be restated as${\displaystyle x\subset y\to (y\subset x\to x=y)}$. The definition D8, which defines the symbol${\displaystyle =}$ for identity or equality, defines${\displaystyle (\alpha =\beta )}$ as shorthand for${\displaystyle (\gamma )((\alpha \in \gamma )\supset (\beta \in \gamma ))}$.^[17] In hisMathematical Logic (1951), having already developedquasi-quotation, Quine defines${\displaystyle ⌜\zeta =\eta ⌝}$ as shorthand for${\displaystyle ⌜(\alpha )(\alpha \in \zeta .\equiv .\alpha \in \eta )⌝}$ (definition D10), and does not define an axiom or principle "of extensionality" at all.^[18]

Thomas Forster, however, ignores these fine distinctions, and considers NF to accept the axiom of extensionality in its ZF form.^[19]

In theScott–Potter (ZU) set theory, the "extensionality principle"${\displaystyle (\mathrm{\forall }x)(x\in a\iff x\in b)\Rightarrow a=b}$ is given as a theorem rather than an axiom, which is proved from the definition of a "collection".^[20]

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Anur-element 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 differentlogical type from sets; in this case,${\displaystyle B\in A}$ makes no sense if${\displaystyle A}$ is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require${\displaystyle B\in A}$ to be false whenever${\displaystyle A}$ is an ur-element.In this case, the usual axiom of extensionality would then imply that every ur-element is equal to theempty set.To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

${\displaystyle \mathrm{\forall }A\mathrm{\forall }B(\mathrm{\exists }X(X\in A)⟹[\mathrm{\forall }Y(Y\in A⟺Y\in B)⟹A=B]).}$

That is:

Given any setA and any setB,ifA is a nonempty set (that is, if there exists a memberX ofA),then ifA andB have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to define${\displaystyle A}$ itself to be the only element of${\displaystyle A}$whenever${\displaystyle A}$ is an ur-element. While this approach can serve to preserve the axiom of extensionality, theaxiom of regularity will need an adjustment instead.

• Extensionality
• Extensional context
• Extension (predicate logic)
• Set theory
• Glossary of set theory

• Axioms of set theory
• Urelements

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