An ordinal analysis for theories of self-referential truth.pdf

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1、Arch.Math.Logic(2010)49:213247DOI10.1007/s00153-009-0170-2MathematicalLogicAnordinalanalysisfortheoriesofself-referentialtruthGrahamEmilLeigh·MichaelRathjenReceived:18November2009/Accepted:23November2009/Publishedonline:12January2010©Springer-Verlag2010Abstra

2、ctThefirstattemptatasystematicapproachtoaxiomatictheoriesoftruthwasundertakenbyFriedmanandSheard(AnnPureApplLog33:121,1987).Theretwelveprinciplesconsistingofaxioms,axiomschemataandrulesofinference,eachembodyingareasonablepropertyoftruthwereisolatedforstudy.Wor

3、kingwithabasetheoryoftruthconservativeoverPA,FriedmanandSheardraisedthefollowingquestions.WhichsubsetsoftheOptionalAxiomsareconsistentoverthebasetheory?Whataretheproof-theoreticstrengthsoftheconsistenttheories?ThefirstquestionwasansweredcompletelybyFriedmanand

4、Sheard;allsubsetsoftheOptionalAxiomswereclassifiedaseitherconsistentorinconsistentgivingrisetoninemaximalconsis-tenttheoriesoftruth.Theyalsodeterminedtheproof-theoreticstrengthoftwosubsetsoftheOptionalAxioms.TheaimofthispaperistocontinuetheworkbegunbyFried-man

5、andSheard.Wewillestablishtheproof-theoreticstrengthofalltheremainingseventheoriesandrelatetheirarithmeticparttowell-knowntheoriesrangingfromPAtothetheoryof
1dependentchoice.1KeywordsTheoriesoftruth·Relativeconsistencyandinterpretations·Ordinalanalysis·Cutelim

6、inationResearchofthefirstauthorwaspartiallysupportedbyaUniversityofLeedsResearchScholarshipandtheMarieCurieEarlyStageTrainingNetworkMATHLOGAPS(MEST-CT-2004-504029).ResearchofthesecondauthorwassupportedbyaRoyalSocietyInternationalJointProjectsaward2006/R3.These

7、condauthorwouldliketothanktheSwedishCollegiumforAdvancedStudyinUppsalaforprovidinganexcellentresearchenvironmentforthecompletionofthispaper.G.E.Leigh(B)·M.RathjenDepartmentofPureMathematics,UniversityofLeeds,LeedsLS29JT,UKe-mail:grahaml@maths.leeds.ac.ukM.Rat

8、hjene-mail:rathjen@maths.leeds.ac.uk123214G.E.Leigh,M.RathjenMathematicsSubjectClassification(2000)03F03·03F25·03F05·03B421IntroductionInthefaceoftheparadoxes,therearethreepossibleroutesth