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The Path to the Iterative Conception

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The Path to the Iterative Conception SetSet theorytheory isis thethe branchbranch ofof mathematicsmathematics concernedconcerned withwith collections.collections. ItIt isis alsoalso aa principalprincipal subjectsubject inin thethe philosophyphilosophy ofof mathematicsmathematics duedue toto itsits uniqueunique positionposition asas TheThe IterativeIterative thethe ““foundationfoundation ofof mathematicsmathematics””.. TheThe philosophicalphilosophical statusstatus ofof “set”“set” isis aa centralcentral issueissue ofof debate,debate, andand wewe focusfocus onon oneone majormajor CC o o n n c c e e p p t t i i o o n n view,view, thethe iterativeiterative conceptionconception ofof set.set. InIn hishis Grundgesetze,Grundgesetze, thethe inventorinventor ofof oo f f SS e e t t modernmodern logic,logic, GottlobGottlob Frege,Frege, advancedadvanced asas hishis ‘‘BasicBasic LawLaw VV’’ aa principleprinciple incorporatingincorporating thethe compellingcompelling ideaidea thatthat forfor anyany things,things, somesome setset hashas themthem asas itsits elements.elements. Unfortunately,Unfortunately, oneone ofof thethe setssets postulatedpostulated byby thethe theory,theory, thethe collectioncollection ofof allall ‘‘non-self-non-self- DueDue toto thethe paradoxes,paradoxes, wewe areare concernedconcerned toto memberedmembered’’ sets,sets, waswas shownshown toto generategenerate anan demonstratedemonstrate thethe validityvalidity ofof ourour intuitiveintuitive inconsistencyinconsistency byby thethe BritishBritish logicianlogician mathematicalmathematical reasoning.reasoning. WeWe needneed toto bebe BertrandBertrand Russell.Russell. TheThe dreamdream ofof aa safesafe explicitexplicit aboutabout thethe principlesprinciples andand rulesrules wewe foundationfoundation forfor thethe 'queen'queen ofof sciences'sciences' use.use. receivedreceived aa hardhard blow.blow. SetSet theorytheory itselfitself waswas contradictory.contradictory. SuitablySuitably specifiedspecified principlesprinciples andand rulesrules ofof mathematicsmathematics areare calledcalled anan axiomaticaxiomatic system.system. TheThe standardstandard axiomaticaxiomatic systemsystem forfor setset theorytheory isis knownknown asas Zermelo-FraenkelZermelo-Fraenkel setset theory,theory, oror ZF.ZF. ThisThis theorytheory hashas thethe powerpower toto recoverrecover allall ofof thethe mathematicsmathematics thatthat hadhad beenbeen developeddeveloped inin thethe centuriescenturies beforebefore itsits AnAn influentialinfluential picturepicture ofof thethe setset theoretictheoretic discovery.discovery. universeuniverse describesdescribes setssets asas ‘formed’‘formed’ inin anan iteratediterated seriesseries ofof stages.stages. 3. Carry on in this way through the finite and transfinite stages: ∅ ∅ ∅ {{{{∅ }} }} {{∅ , , {{ ∅ }} }} 3, 4, 5, …, ω, ω+1, OurOur projectproject aimsaims toto enrichenrich thethe languagelanguage ofof Key. … setset theorytheory withwith newnew expressiveexpressive resourcesresources inin ∅: The {}{}∅∅ 2. Sets of all sets orderorder toto reconcilereconcile ourour pre-theoreticpre-theoretic empty set formed at stages formed at stages conceptionconception ofof setset withwith thethe powerpower ofof modernmodern {a,b}: The set 0 and 1 are with elements formed setset theory.theory. a and b ∅∅ 1. Sets of all WithWith suitablesuitable modality,modality, wewe cancan faithfullyfaithfully 0. Start with sets formed at formaliseformalise thethe guidingguiding ideaidea inin thethe naïvenaïve nothing. stage 0 are picturepicture thatthat anyany setssets cancan formform aa set.set. ThisThis formed formed allowsallows forfor aa naturalnatural butbut consistentconsistent axiomatisationaxiomatisation ofof thethe iterativeiterative conception,conception, whichwhich allowsallows usus toto recoverrecover aa greatgreat dealdeal ofof ZF.ZF. 1971.1971. GeorgeGeorge BoolosBoolos publishespublishes 'The'The IterativeIterative ConceptionConception ofof Set'Set' TheThe pathpath toto thethe iterativeiterative conception...conception... 1922.1922. AbrahamAbraham FraenkelFraenkel extendsextends Zermelo'sZermelo's axiomsaxioms toto createcreate ZFZF 1908.1908. ErnstErnst ZermeloZermelo providesprovides thethe firstfirst axiomatisationaxiomatisation ofof setset theorytheory 1902.1902. BertrandBertrand RussellRussell writeswrites toto Frege,Frege, outliningoutlining thethe discoverydiscovery ofof thethe paradoxparadox inin Frege'sFrege's systemsystem 1893.1893. GottlobGottlob FregeFrege publishespublishes thethe ill-fatedill-fated GrundgesetzeGrundgesetze derder ArithmetikArithmetik,, vol.vol. II 1874.1874. GeorgGeorg CantorCantor discoversdiscovers infinityinfinity comescomes inin differentdifferent sizes.sizes. SetSet theorytheory isis bornborn Kentaro Fujimoto, Graham Leigh, Carlo Nicolai, James Studd Faculty of Philosophy, University of Oxford Background image: selections from Ernst Zermelo's 'Untersuchungen uber die Grundlagen der Mengenlehre'. Mathematische Annalen, 65 (1908), pp. 261-281. [email protected] http://users.ox.ac.uk/~reflect.
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