SetSet theory theory is is the the branch branch of of mathematics mathematics concernedconcerned with with collections. collections. It It is is also also a a principalprincipal subject subject in in the the philosophy philosophy of of mathematicsmathematics due due to to its its unique unique position position as as TheThe IterativeIterative thethe ““foundationfoundation of of mathematics mathematics””.. The The philosophicalphilosophical status status of of “set” “set” is is a a central central 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 his his Grundgesetze,Grundgesetze, thethe inventor inventor of of oo f f S S e e t t modernmodern logic, logic, Gottlob Gottlob Frege, Frege, advanced advanced as as hishis ‘‘BasicBasic LawLaw VV’’ aa principleprinciple incorporatingincorporating thethe compelling compelling idea idea that that for for any any things, things, somesome setset hashas themthem asas itsits elements.elements.

Unfortunately,Unfortunately, oneone ofof thethe setssets postulatedpostulated byby thethe theory, theory, the the collection collection of of all all ‘‘non-self-non-self- DueDue to to the the paradoxes, paradoxes, we we are are concerned concerned to to memberedmembered’’ sets,sets, waswas shownshown toto generategenerate anan demonstratedemonstrate the the validity validity of of our our intuitive intuitive inconsistencyinconsistency by by the the British British logician logician mathematicalmathematical reasoning. reasoning. We We need need to to be be BertrandBertrand Russell. Russell. The The dream dream of of a a safe safe explicitexplicit about about the the principles principles and and rules rules we we foundationfoundation for for the the 'queen 'queen of of sciences' sciences' use.use. receivedreceived aa hardhard blow.blow. SetSet theorytheory itselfitself waswas contradictory.contradictory. SuitablySuitably specified specified principles principles and and rules rules of of mathematicsmathematics areare calledcalled anan axiomaticaxiomatic system.system. The The standard standard axiomatic axiomatic system system for for set set theorytheory is is known known as as Zermelo-Fraenkel Zermelo-Fraenkel set set theory,theory, oror ZF.ZF. ThisThis theorytheory hashas thethe powerpower toto recoverrecover allall ofof thethe mathematicsmathematics thatthat hadhad beenbeen developeddeveloped in in the the centuries centuries before before its its 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 project project aims aims to to enrich enrich the the language language of of Key. … setset theorytheory withwith newnew expressiveexpressive resourcesresources inin ∅: The {}{}∅∅ 2. Sets of all sets orderorder to to reconcile reconcile our our pre-theoretic pre-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 suitable suitable modality, modality, we we can can faithfully faithfully 0. Start with sets formed at formaliseformalise the the guiding guiding idea idea in in the the naïve naïve nothing. stage 0 are picturepicture that that any any sets sets cancan form form a a set. set. This This formed formed allowsallows for for a a natural natural but but consistent consistent axiomatisationaxiomatisation of of the the iterative iterative conception, conception, whichwhich allows allows us us to to recover recover a a great great deal deal of of 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