ZFC: Why? What? And, how?. Naïve set theory ...

That's it! Zermelo-Fraenkel set theory with the axiom of choice, ZFC, consists of the 10 axioms we just learned about: extensionality, empty set ... SigninLatestMostPopularAllstoriesAboutWriteforCantor'sParadisePrivatdozentNewsletterZFC:Why?What?And,how?RobertPassmannFollowMay20·11minreadZermelo-Fraenkelsettheorywiththeaxiomofchoiceisconsideredthestandardfoundationformathematics.Butwhy?Whatareitsaxioms?Andhowdoesthistheoryallowustosettletheparadoxesofnaïvesettheory?Exercisesinsettheory.(Photobyauthor)Naïvesettheorydoesn’tdothejobOurjourneybeginswithnaïvesettheory.That’snotaformallogicaltheoryaxiomatisedinaformallanguage.Rather,naïvesettheoryisaninformalcollectionofassumptionsaboutsets,formulatedinnaturallanguage:Foranytwosetsthere’saunionandanintersection.Wehaveasetofnaturalnumbers,andfromthosewecanconstructtherealnumbers.Thecrucialassumptionofnaïvesettheory,however,istheso-calledunrestrictedcomprehensionschemastatingthat:ForanypropertyP(x),thereisasetconsistingofexactlythosexthatsatisfyP.Atfirstlook,comprehensioncertainlymakessense:Givenanypropertyweshouldbeabletotalkaboutthesetofallthoseobjectssatisfyingtheproperty.Forexample,whenwetalkaboutthesetofprimenumbers—thosexthatarenaturalnumbersandhaveexactlytwodivisors.Itturnsout,however,thattheunrestrictedcomprehensionschemaishighlyproblematic.BetrandRusselldiscoveredthefollowingparadoxin1901—Russell’sparadox:TakeP(x)tobethepropertythat‘xdoesnotcontainitself,’or‘x∉x’insymbolicnotation.Bytheunrestrictedcomprehensionschema,theremustbeasetyconsistingofallthosesetsxthatsatisfyP(x).Thatis,yconsistsofallsetsthatdonotcontainthemselves.Doesycontainitself?Isitthecasethaty∈y?Ifso,thenP(y)musthold.Buttheny∉y.That’sacontradiction.Itmustthereforebethecasethaty∉y.ButthismeansthatP(y)cannothold.Thisgetsusthaty∈y.Yetanothercontradiction.Russell’sparadoxthereforeshowsthattheunrestrictedcomprehensionschemaisinconsistent.Ifweassumeit,thenweintroducepropercontradictionsintoourreasoning.Moreset-theoreticparadoxeswerefoundaroundtheturnofthe20thcentury.AnotherexampleisBurali-Forti’sparadox:Heusedtheunrestrictedcomprehensionschematodefinethesetzofallordinals.Thatis,hechoseP(x)tomeanthat‘xisanordinal’andthenappliedcomprehensiontogetthesetzofallsetssatisfyingP(x).Itthenturnsoutthatzitselfsatisfiesthedefinitionofanordinal.Soz∈z,andthereforez