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Infinite vs unbounded

发布时间：2023-02-22 18:34:18 来源：文档文库

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InfinityinEthicsRoutledgeEncyclopediaofPhilosophy,ElectronicSupplement,Forthcoming2001Puzzlescanariseinethicaltheory(aswellasdecisiontheorywheninfinityisinvolved.Thepuzzlesariseprimarilyintheories—suchasconsequentialisttheories—thatappealtothevalueofactionsorstatesofaffairs.Section1addressesthequestionofwhetheronesourceofvalue(suchasmajoraestheticpleasurescanbeinfinitelymorevaluablethananother(suchasminorgustatorypleasures.Anaffirmativeanswerisgivenbyappealingtothenotionoflexicographicpriority.Section2addressthequestionofwhatmoralityrequireswhenthereareaninfinitenumberoffeasibleoptionsandnooptionismaximallyvaluable?Insuchcases,itissuggested,moralitycandemandnomorethanthatwe“almostmaximize”or(moreweaklythatwe“satisfice”.Section3addressesapuzzlethatcanarisewhentimeisinfinitelylong.Isastateofaffairswithtwounitsofvalueateachtimemorevaluablethanastateofaffairswithoneunitateachtime(eventhoughbothproduceinfiniteamountsofvalue?Aplausibleprincipleisintroducedthatanswersaffirmatively,butitfacescertainproblems.Section4addressesapuzzlethatcanarisewhentimeisfinitebutinfinitelydivisible.1.LexicographicPriorityCanthevalueofsomeeventsorstates(e.g.,thepleasureofhearingmusicthatoneabsolutelylovesbeinfinitelygreaterinrelativetermsthanthevalueofsomeother
eventorstate(e.g.,thepleasureeatingacarrotthatoneisnotespeciallyexcitedabout?Isitpossible,thatis,thatthereisnofinitenumberofthelattereventssuchthatthevalueofallthoseeventstogetherisatleastasgreatasthevalueoftheformerevent?Thisisimpossible,ifallstatesandeventshavesomestandardfinitevalue.Onecan,however,coherentlyrejectthisassumption.First,valueneednotberepresentablebynumbers.Itmaysimplybeordinallyrepresentablebyarankingrelation(i.e.,asmore,less,orequallyvaluable;butnoassignmentofspecificnumbersforvalue.Allelsebeingequal,moreoftheinfinitelylessvaluablesourcesofvaluemakestheworldmorevaluable,butnofinitenumberofsuchsourcescanevercompensateforthelossofoneoftheformersourcesofvalue.Theinfinitelymorevaluablesourcesofvaluearesimplylexicographicallypriorinthegenerationofoverallvaluetotheinfinitelylessvaluablesourcesofvalue.Thus,wecanmakeperfectsenseoftheidea,ifwedonotrequirethatvaluebenumericallyrepresentable.Second,evenifonerequiresthatnumbersbeassignedtothevalueofstatesoftheworld,thiscanbedoneusinginfinitesimalnumbersofnon-standardarithmetic.Instandardmathematics,therearenonumbersthatareinfinitesimallysmall.Inthe1960s,however,AbrahamRobinson,amathematician,provedthatonecanmakeperfectmathematicalsenseofsuchinfinitesimals,andthusthatitislegitimatetopositthem.Theadditionofapositiveinfinitesimaltoagivennumberproducesalargernumber,butthesumoffinitelymanyinfinitesimalsstillisstillinfinitesimallysmall,andhencesmallerthananyfinitenumber(althoughgreaterthaneachoftheoriginal2

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