axiom of extensionality

The term extensionality, as used in 'Axiom of Extensionality' has its roots in logic. An intensional definition describes the necessary and sufficient conditions for a term to apply to an object. For example: "An even number is an integer which is divisible 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, the extension of a predicate is the set of all things for which the predicate is true.{{Cite book |last=Roy T Cook |url=https://archive.org/details/roy-t.-cook-a-dictionary-of-philosophical-logic/page/155/mode/2up?q=%22INTENSIONAL+DEFINITION%22 |title=A Dictionary Of Philosophical Logic |date=2010 |isbn=978-0-7486-2559-8 |pages=155}}

The logical term was introduced to set theory in 1893, Gottlob Frege attempted to use this idea of an extension formally in his Basic Laws of Arithmetic (German: Grundgesetze der Arithmetik),{{Cite book |last=Lévy |first=Azriel |url=https://archive.org/details/basicsettheory00levy_0/mode/2up?q=Frege |title=Basic set theory |date=1979 |publisher=Berlin ; New York : Springer-Verlag |isbn=978-0-387-08417-6 |pages=5}}{{Cite book |last=Frege |first=Gottlob |url=https://archive.org/details/bub_gb_LZ5tAAAAMAAJ/page/n105/ |title=Grundgesetze der arithmetik |date=1893 |publisher=Jena, H. Pohle |pages=69}} where, if $F$ is a predicate, its extension $\backslash varepsilon\; F$, is the set of all objects satisfying $F$.{{Citation |last=Zalta |first=Edward N. |title=Frege's Theorem and Foundations for Arithmetic |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/frege-theorem/ |access-date=2025-01-16 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}} For example if $F(x)$ is "x is even" then $\backslash varepsilon\; F$ is the set $\backslash \{\; \backslash cdots\; ,\; -4,\; -2,\; 0,\; 2,\; 4,\; \backslash cdots\; \backslash \}$. In his work, he defined his infamous Basic Law V as:{{Sfn|Ferreirós|2007|p=304}}$$\backslash varepsilon\; F\; =\; \backslash varepsilon\; G\; \backslash equiv\; \backslash forall\; x\; (F(x)\; \backslash equiv\; G(x)\; )$$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 to Russell's paradox. The first explicit statement of the modern Axiom of Extensionality was in 1908 by Ernst Zermelo in a paper on the well-ordering theorem, where he presented the first axiomatic set theory, now called Zermelo set theory, which became the basis of modern set theories.{{Citation |last=Hallett |first=Michael |title=Zermelo's Axiomatization of Set Theory |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/zermelo-set-theory/ |access-date=2025-01-16 |edition=Fall 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}} 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,Oxford English Dictionary, s.v. “Extensionality (n.)” December 2024 particularly with the formalization of logic and set theory by figures like Alfred Tarski and John von Neumann.

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

:$\backslash forall\; x\backslash forall\; y\; \backslash ,\; [\backslash forall\; z\; \backslash ,\; (\backslash left.z\; \backslash in\; x\backslash right.\; \backslash leftrightarrow\; \backslash left.\; z\; \backslash in\; y\backslash right.)\; \backslash rightarrow\; x=y]${{Cite web |title=Set Theory > Zermelo-Fraenkel Set Theory (ZF) (Stanford Encyclopedia of Philosophy) |url=https://plato.stanford.edu/entries/set-theory/ZF.html |access-date=2024-11-24 |website=plato.stanford.edu |language=en}}{{Cite web |title=Zermelo-Fraenkel Set Theory |url=https://www.cs.odu.edu/~toida/nerzic/content/set/ZFC.html |access-date=2024-11-24 |website=www.cs.odu.edu}}{{Cite web |title=Naive Set Theory |url=https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/sets/sets.html |access-date=2024-11-24 |website=sites.pitt.edu}}

or in words:

:If the sets $x$ and $y$ have the same members, then they are the same set.

In {{glossary link|pure set theory|glossary=Glossary of set theory}}, all members of sets are themselves sets, but not in set theory with urelements. The axiom's usefulness can be seen from the fact that, if one accepts that $\backslash exists\; A\; \backslash ,\; \backslash forall\; x\; \backslash ,\; (x\; \backslash in\; A\; \backslash iff\; \backslash Phi(x))$, where $A$ is a set and $\backslash Phi(x)$ is a formula that $x$ occurs free in but $A$ doesn't, then the axiom assures that there is a unique set $A$ whose members are precisely whatever objects (urelements or sets, as the case may be) satisfy the formula $\backslash Phi(x)$.

The converse of the axiom, $\backslash forall\; x\backslash forall\; y\; \backslash ,\; [x=y\; \backslash rightarrow\; \backslash forall\; z\; \backslash ,\; (\backslash left.z\; \backslash in\; x\backslash right.\; \backslash leftrightarrow\; \backslash left.\; z\; \backslash in\; y\backslash right.)]$, follows from the substitution property of equality. Despite this, the axiom is sometimes given directly as a biconditional, i.e., as $\backslash forall\; x\backslash forall\; y\; \backslash ,\; [\backslash forall\; z\; \backslash ,\; (\backslash left.z\; \backslash in\; x\backslash right.\; \backslash leftrightarrow\; \backslash left.\; z\; \backslash in\; y\backslash right.)\; \backslash leftrightarrow\; x=y]$.

Quine's New Foundations (NF) set theory, in Quine's original presentations of it, treats the symbol $=$ 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, $x=y$ is treated as shorthand either for $\backslash forall\; z\; \backslash ,\; \backslash left.(x\; \backslash in\; z\backslash right.\; \backslash rightarrow\; \backslash left.\; y\backslash in\; z\backslash right.)$, as in the original 1937 paper, or for $\backslash forall\; z\; \backslash ,\; \backslash left.(z\; \backslash in\; x\backslash right.\; \backslash leftrightarrow\; \backslash left.\; z\backslash in\; y\backslash right.)$, as in Quine's Mathematical Logic (1951). The second version of the definition is exactly equivalent to the antecedent 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 $=$ 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's New Foundations for Mathematical Logic (1937), the original paper of NF, the name "principle of extensionality" is given to the postulate P1, $(\; (\; x\; \backslash subset\; y\; )\; \backslash supset\; (\; (\; y\; \backslash subset\; x\; )\; \backslash supset\; (x\; =\; y)\; )\; )$,{{Cite journal |last=Quine |first=W. V. |date=1937 |title=New Foundations for Mathematical Logic |url=https://www.jstor.org/stable/2300564 |journal=The American Mathematical Monthly |volume=44 |issue=2 |pages=74, 77 |doi=10.2307/2300564 |jstor=2300564 |issn=0002-9890}} which, for readability, may be restated as $x\; \backslash subset\; y\; \backslash rightarrow\; (\; y\; \backslash subset\; x\; \backslash rightarrow\; x\; =\; y\; )$. The definition D8, which defines the symbol $=$ for identity or equality, defines $(\backslash alpha\; =\; \backslash beta)$ as shorthand for $(\backslash gamma)\; \backslash ,\; (\backslash left.(\backslash alpha\; \backslash in\; \backslash gamma\backslash right.)\; \backslash supset\; (\backslash left.\; \backslash beta\backslash in\; \backslash gamma\backslash right.))$. In his Mathematical Logic (1951), having already developed quasi-quotation, Quine defines $\backslash ulcorner\; \backslash zeta=\backslash eta\; \backslash urcorner$ as shorthand for $\backslash ulcorner\; (\backslash alpha)\; \backslash ,\; (\backslash left.\backslash alpha\; \backslash in\; \backslash zeta\backslash right.\; \backslash ;\; .\; \backslash equiv\; \backslash ,\; .\; \backslash ,\; \backslash left.\; \backslash alpha\; \backslash in\; \backslash eta\backslash right.)\; \backslash urcorner$ (definition D10), and does not define an axiom or principle "of extensionality" at all.{{Cite journal |last=Quine |first=W. V. |date=1951-12-31 |title=Mathematical Logic |url=http://dx.doi.org/10.4159/9780674042469 |journal=DeGruyter |pages=134–136 |doi=10.4159/9780674042469|isbn=978-0-674-04246-9 }}

Thomas Forster, however, ignores these fine distinctions, and considers NF to accept the axiom of extensionality in its ZF form.{{Citation |last=Forster |first=Thomas |title=Quine's New Foundations |date=2019 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/quine-nf/ |access-date=2024-11-24 |edition=Summer 2019 |publisher=Metaphysics Research Lab, Stanford University}}

In the Scott–Potter (ZU) set theory, the "extensionality principle" $(\; \backslash forall\; x\; )\; (\backslash left.x\; \backslash in\; a\backslash right.\; \backslash Leftrightarrow\; \backslash left.\; x\; \backslash in\; b\backslash right.)\; \backslash Rightarrow\; a=b$ is given as a theorem rather than an axiom, which is proved from the definition of a "collection".{{Cite book |last=Potter |first=Michael D. |url=https://www.worldcat.org/title/ocm53392572 |title=Set theory and its philosophy: a critical introduction |date=2004 |publisher=Oxford University Press |isbn=978-0-19-926973-0 |location=Oxford; New York |pages=31 |oclc=ocm53392572}}

{{Unreferenced section|date=November 2024}}

An ur-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 different logical type from sets; in this case, $B\; \backslash in\; A$ makes no sense if $A$ is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require $B\; \backslash in\; A$ to be false whenever $A$ 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.

To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

:$\backslash forall\; A\; \backslash ,\; \backslash forall\; B\; \backslash ,\; (\; \backslash exists\; X\; \backslash ,\; (X\; \backslash in\; A)\; \backslash implies\; [\; \backslash forall\; Y\; \backslash ,\; (Y\; \backslash in\; A\; \backslash iff\; Y\; \backslash in\; B)\; \backslash implies\; A\; =\; B\; ]\; \backslash ,\; ).$

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 $A$ itself to be the only element of $A$

whenever $A$ is an ur-element. While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

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

{{refbegin}}

• {{Citation |last=Ferreirós |first=José |title=Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought |year=2007 |edition=2nd revised |publisher=Birkhäuser |isbn=978-3-7643-8349-7}}
• 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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Category:Axioms of set theory

Category:Urelements