Editing Axiom schema of specification (section)

{{short description|Concept in axiomatic set theory}}
{{redirect|Axiom of separation|the separation axioms in topology|separation axiom}}

In many popular versions of [[axiomatic set theory]], the '''axiom schema of specification''',<ref name=":1">{{Cite web |title=AxiomaticSetTheory |url=https://www.cs.yale.edu/homes/aspnes/pinewiki/AxiomaticSetTheory.html |access-date=2024-06-08 |website=www.cs.yale.edu |at=Axiom Schema of Specification}}</ref> also known as the '''axiom schema of separation''' (''Aussonderungsaxiom''),<ref name="SuppesAxiomatic">{{Cite book |last=Suppes |first=Patrick |url=https://books.google.com/books?id=sxr4LrgJGeAC |title=Axiomatic Set Theory |date=1972-01-01 |publisher=Courier Corporation |isbn=978-0-486-61630-8 |pages=6,19,21,237 |language=en |quote=}}</ref> '''subset axiom<ref name=":0">{{Cite book |last=Cunningham |first=Daniel W. |title=Set theory: a first course |date=2016 |publisher=Cambridge University Press |isbn=978-1-107-12032-7 |series=Cambridge mathematical textbooks |location=New York, NY |pages=22,24-25,29}}</ref>''', '''axiom of class construction''',<ref>{{Cite book |last=Pinter |first=Charles C. |url=https://books.google.com/books?id=iUT_AwAAQBAJ |title=A Book of Set Theory |date=2014-06-01 |publisher=Courier Corporation |isbn=978-0-486-79549-2 |pages=27 |language=en}}</ref> or '''axiom schema of restricted comprehension''' is an [[axiom schema]].  Essentially, it says that any definable [[subclass (set theory)|subclass]] of a set is a set.

Some mathematicians call it the '''axiom schema of comprehension''', although others use that term for '''''unrestricted'' comprehension''', discussed below.

Because restricting comprehension avoided [[Russell's paradox]], several mathematicians including [[Ernst Zermelo|Zermelo]], [[Abraham Fraenkel|Fraenkel]], and [[Gödel]] considered it the most important axiom of set theory.<ref name="Ebbinghaus2007">{{cite book|author=Heinz-Dieter Ebbinghaus|title=Ernst Zermelo: An Approach to His Life and Work|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-49553-6|page=88}}</ref>