Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?

Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?

InformallytheideaofthisquestionisaboutwhethertherulesofsettheorycanbederivedasatransferofsomerulesfromthehereditarilyfinitesetrealmandwhetherthistransferprincipleitselfcanbecoinedfornotionsotherthanthefinitenotionTheprincipleIwanttonegotiateisifphiisapropertythatisdefinablebyaformulainthelanguageofsettheorythatisstrictlyshorterthantheshortestparameterfreeformulainthatlanguagethatcandefinefinitenessthenifphiisCLOSEDonthethehereditarilyfinitesetworldthenitcanbegeneralizedoverthewholerealmofsetsThecrudeinformalideaisthatifapropertythatcannotmentionfinitenessgeneralizesoverthewholehereditarilyfinitesetrealmthenitcangobeyonditToformallycapturethatIllworkupinaclasstheorysowedefinesetasanelementofaclassthelanguageofthetheoryismonosortedfirstorderlogicwithidentityandmembershipwestipulateaxiomsofExtensionalityasinZFClasscomprehensionschemaforallx1xnexistsxxysetywedgephiyx1xnTheemptyclassisasetSingletonsforallxsetxtosetxBooleanUnionforallxysetxwedgesetytosetxcupyDefinefinAiffforallKforallxxinAtoexistsyyinKwedgexinywedgeforallzzinytozxwedgeforallabainKwedgebinKtoexistsccinKwedgeforallddincleftrightarrowdinalordinbtoAinKInEnglishAisfiniteifandonlyifitisanelementofeveryclassKthatisclosedunderBooleanunionandthathasthesingletonsofallelementsofAamongitselementsIthinkthisisalongtheshortestwaytodefinefinitesetinthefirstorderlanguageofsettheoryPerhapstheaboveformulacanbeshortenedfurtherorperhapsthereisanothershorterparameterfreeformulationofxisafinitesetinthelanguageofsettheoryhoweverforthesakeofpresentationherewelltakethisformulatobetheshortestformuladefiningfinitenessforallxxtextishereditarilyfinitetosetxWherexishereditarilyfiniteisdefinedasthetransitiveclosureclassofxbeingfiniteWeshalldenotetheclassofallsetsbyVandtheclassofallhereditarilyfintiesetsbyHFHFinVTheprincipleofTransferfromthepurefiniteworldifphiyxisaformulashorterthananyformuladefiningfinitenessinwhichonlysymbolsyxoccurfreeandthoseonlyoccurfreethenforallxxinHFtoforallyphiyxtoyinHFtoforallxinVforallyphiyxtoyinVNowitisclearthatallaxiomsofUnionPowerandSeparationoversetsarederivablefromtheabovetransferprincipleandsoZCisinterpretablehereActaullyifwerestrictphitohavenomorethanthreeatomicsubformulaswecanstillinterpretthewholeofZCReplacementisnotinterpretablebythisprincipleYetaminormodificationofthisprinciplemightsucceedinprovingreplacementoversetsthiscanbedonebychangingtheclosurepropertytoinvolveonlysubsetsofHFwhatIcallasproximityclosureoverHFsotorestatethat8TheprincipleofTransferfromproximityofthepurefiniteworldifphiyxisaformulashorterthananyformuladefiningfinitenessinwhichonlysymbolsyxoccurfreeandthoseonlyoccurfreethenforallxxinHFtoforallysubseteqHFphiyxtoyinHFtoforallxinVforallyphiyxtoyinVThatreplacementisprovablecanbeshownfromexaminingthefollowingformulawhoselengthisshorterthananyformuladefiningfinitenessexistsFforallmminFtoexistsabainAwedgebinBwedgeainmwedgebinmwedgeforallmnminFwedgeninFwedgeexistskkinmwedgekinntonmNowifAishereditarilyfiniteandBisasubsetofHFthatfulfillstheaboveformulathenBishereditarilyfinitethismeanthatthepropertydefinedbytheaboveformulaisproximityclosedoverthehereditarilyfiniteworldIdonthaveanyproofofconsistencyoftheseprinciplesbutifthereisnoclearinconsistencyofthoserelativetoZForMKorsomeextensionofthosethencoulditbepossibletothinkofextendingthatprincipletopropertiesotherthanxisfinitesowegeneralizeittosomelinepropertiessoforapropertyPinthatlinewestipulatethatanypredicateQthatisclosedoverthepurePworldwouldgeneralizeoverthewholesetworldorevenstrongeranypredicateQthatisproximityclosedoverthepurePworldwouldgeneralizeoverthewholesetworldOfcourseinbothcasesQmustbeexpressiblebyaformulastrictlyshorterthantheshortestexpressiondefiningpropertyPandalsowestipulateparallelaxiomssufficienttodefinethepropertyPalsoaxiomsassertingtheelementhoodofallhereditarilyPclassesandtheexistenceofasetofallhereditarilyPsetsOfcoursethiscanonlybedoneforsomeselectedlineofpropertiesisthatpossibleoritisinvolvedwithclearinconsistenciesandwhatwouldbethegeneralqualificationofsuchpropertyP

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