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