Editing Axiom of extensionality (section)

== Etymology ==
The term ''[[extensionality]]'', as used in ''<nowiki/>'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 (logic)|extension]] of a [[Predicate (mathematical logic)|predicate]] is the set of all things for which the predicate is true.<ref>{{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}}</ref>

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|''Basic Laws of Arithmetic'']] (German: ''Grundgesetze der Arithmetik''),<ref>{{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}}</ref><ref>{{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}}</ref> where, if <math>F</math> is a predicate, its extension (German: ''Umfang'') <math>\varepsilon F</math>, is the set of all objects satisfying <math>F</math>.<ref>{{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}}</ref> For example if <math>F(x)</math> is "x is even" then <math>\varepsilon F</math> is the set <math>\{ \cdots , -4, -2, 0, 2, 4, \cdots \} </math>. In his work, he defined his infamous ''[[Basic Law V]]'' as:{{Sfn|Ferreirós|2007|p=304}}<math display="block">\varepsilon F = \varepsilon G \equiv \forall x (F(x) \equiv G(x) ) </math>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.<ref>{{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}}</ref> 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,<ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/1191853349|Extensionality (n.)]]” December 2024</ref> particularly with the formalization of logic and set theory by figures like [[Alfred Tarski]] and [[John von Neumann]].