Computational Logic and Set Theory: Applying Formalized Logic to Analysis PDF
Preview Computational Logic and Set Theory: Applying Formalized Logic to Analysis
Computational Logic and Set Theory Jacob T. Schwartz (cid:2) Domenico Cantone (cid:2) Eugenio G. Omodeo Computational Logic and Set Theory Applying Formalized Logic to Analysis Foreword by Martin Davis Prof.Dr.JacobT.Schwartz Prof.EugenioG.Omodeo (January9,1930–March2,2009) Dept.ofMathematics&ComputerScience NewYorkUniversity UniversityofTrieste NewYork,NY ViaValerio12/1 USA 34127Trieste, Italy Prof.DomenicoCantone [email protected] Dept.ofMathematics&ComputerScience UniversityofCatania VialeAndreaDoria6 95125Catania Italy [email protected] ISBN978-0-85729-807-2 e-ISBN978-0-85729-808-9 DOI10.1007/978-0-85729-808-9 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2011934034 ©Springer-VerlagLondonLimited2011 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Coverdesign:VTeXUAB,Lithuania Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) JacobTheodore“Jack”Schwartz(January9,1930–March2,2009),courtesyofDianaRobinson Schwartz Foreword Jack Schwartz, the principal, but alas posthumous, author of this book, turned his seriousattentiontocomputerscienceinthemid1960s.Atthetimehehadalready been recognized as a brilliant young mathematician, and the two volumes of the magisterial Dunford–Schwartz Linear Operators already in print were widely ad- mired.Jacksawthatcomputersweregoingtohavearevolutionaryeffectandthat theexpansionsoftheirusewouldgiverisetomanyfundamentalproblems,andhe wanted to be part of that. He realized that as software became more complex the questionofhowitscorrectnesscouldbeensuredwouldbecomeevermorecritical. Moreoverhesawformallogicembodiedincomputerprogramsasanimportantpart oftheanswer. AsubstantialpartofJack’sresearchprogramincomputersciencederivedfrom hisappreciationofthepossibilityofexpressingmathematicaldiscourseinthelan- guage of set theory. In much the same way that the seventeenth century work of Descartes and Fermat had shown that propositions of Euclid’s geometry could be regarded as statements in the language of algebra, so the twentieth century con- tributionsofRussell,Zermelo,andvonNeumannshowedhowpropositionsofthe variousbranchesofmathematicscouldberegardedasstatementsinthelanguageof settheory.Thisappreciationledhiminthreedirections: 1. He designed SETL, a general purpose high level programming language based onthelanguageofsettheory.Theneedtoachieveacceptableperformancefrom softwarewritteninalanguagethatmadenoconcessionstothevagariesofcom- puterarchitectureledtoworkoncompileroptimizationinfruitfulcollaboration withresearchersfromIBM. 2. After studying Heinrich Behmann’s algorithm for the decision problem of sec- ondordermonadicpredicatecalculus,Jacksawthatthesemethodscouldbeex- tendedtoyieldalgorithmsfordecidablefragmentsofsettheory.Overaperiodof decades,workingwithagroupofcollaboratorsalmostallfromItaly,whowere firststudentsatNewYorkUniversityandthenbecamedistinguishedscientistsin theirownright,asurprisingcollectionofnon-trivialmathematicswasfoundto liewithinthescopeofsuchalgorithms. vii viii Foreword 3. WorkingwithsomeofthesesameItalianresearchers,acomputerprogramwas designed and implemented (at least as a prototype) that could verify the cor- rectness of mathematical proofs presented in the language of set theory. Jack proposedtousethisverifiertocertifythecorrectnessofasubstantialbodyofthe fundamentalsofmathematicalanalysis.Thiswastoincludeproofsofthebasic propertiesoftherealandcomplexnumbersystemsdefinedinset-theoreticterms, thefundamentalpropertiesoflimits,continuityandthedifferentialandintegral calculus, and was to culminate in a proof of the Cauchy Integral Theorem of complexanalysis. Thepresentvolumeisconcernedwiththisverifier,itsuseanditscontext.How- everthiscontextistobeunderstoodinanextremelybroadsense.Someofthework on decidable fragments of set theory is presented in a context that includes other algorithmsfromvarioussourcesforbranchesoflogicaswell.Themainmetamath- ematicaltheoremscoveredinamoderncourseinmathematicallogicarehere:the completenesstheoremandthetwoincompletenesstheoremsofGödel.Suchtopics asreflectionprinciplesandlargecardinalsarehereaswell.ThosefamiliarwithJack Schwartz’s mode of thought and with his way of putting his own stamp on a field willhavenotroublehearinghisvoiceinthisimportantthought-provokingbook. MartinDavis ProfessorEmeritus,CourantInstitute,NewYorkUniversity VisitingScholar,UniversityofCalifornia,Berkeley Preface InJune2000,thethirdnamedauthorvisitedNewYorkUniversityandwasinvited byJackSchwartztoreadwhathecalledthe(“commonshared”)scenario:awide, carefully assembled sequence of definitions, theorems, and proofs, leading from thebarerudimentsofsettheorytothebeginningofmathematicalanalysis.Proofs begantobegappyafterafewhundredpages,andthentotallyabsent,buttheflow ofdefinitionsandtheoremswenton,toculminateinthedefinitionofcomplexline integralandfinallyinthecelebratedCauchyintegraltheoremofcomplexanalysis. Withanimplementationappearingallbutimminent,Jackhadcastasignificant piece of mathematics in rigorous formal detail, honestly asking himself whether a computer program could conceivably process and validate every single step. The resultinglarge-scaleproofscenariowasmeant—inJack’sownwords—“toserveas an essential part of the feasibility study that must precede the developmentof any ambitiousproof-checker”.Eventually,itwouldalsoserveasatesting-benchforthe concreteimplementationoftheproof-checker. Theconceptionofthisbookoncomputationallogicbeganthen.Accordingtoour initialplans,thebookwouldhavedescribedthestructureofaproofverifierrooted insettheoryandwouldalsohavesurveyedatwenty-yearlongstreamofresultson decidablefragmentsofsettheory. Oneyearlater,thesecondnamedauthorvisitedNewYorkinhisturn;athisre- questJackadvancedtheimplementationwork,speedilybringingintoexistencethe proof-checker Referee, also known as Ref, or as ÆtnaNova. This is still a proto- type, but it is reliable and fast enough to give us the possibility of debugging our proofscenarios.Some,thoughnotall,ofthecontentofthisbookisthusrelatedto concreteexperience,whichwearenowpleasedtosharewithourreaders. AverylargeproofscenarioisavailabletodayasaLATEX-generatedPDF-file.But givenitssize(overathousandpages),itseemsappropriatetopublishitontheweb (andeventuallyasaCD)ratherthantoprintit.Asfortheproofverifier,itisusable ontheweb,butitdependsonaSETL2implementation.Sincethereishardlyanyone maintainingtheSETLsystemtoday,Jackundertookwithusare-implementationof theproofverifierinacurrentlymorepopularlanguage.Butthiswilltakesometime; itshouldnotbepermittedtodelaythepublicationofthisbook. ix
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