Andrzej Mostowski and foundational studies PDF
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ANDRZEJ MOSTOWSKI AND FOUNDATIONAL STUDIES This page intentionally left blank Andrzej Mostowski and Foundational Studies Edited by A. Ehrenfeucht DepartmentofComputerScience,UniversityofColorado,CO,USA V.W. Marek DepartmentofComputerScience,UniversityofKentucky,KY,USA and M. Srebrny InstituteofComputerScience,PolishAcademyofSciences,Poland Amsterdam•Berlin•Oxford•Tokyo•Washington,DC ©2008TheauthorsandIOSPress. Allrightsreserved.Nopartofthisbookmaybereproduced,storedinaretrievalsystem, ortransmitted,inanyformorbyanymeans,withoutpriorwrittenpermissionfromthepublisher. ISBN978-1-58603-782-6 LibraryofCongressControlNumber:2007939570 Publisher IOSPress NieuweHemweg6B 1013BGAmsterdam Netherlands fax:+31206870019 e-mail:[email protected] DistributorintheUKandIreland DistributorintheUSAandCanada GazelleBooksServicesLtd. IOSPress,Inc. WhiteCrossMills 4502RachaelManorDrive Hightown Fairfax,VA22032 LancasterLA14XS USA UnitedKingdom fax:+17033233668 fax:+44152463232 e-mail:[email protected] e-mail:[email protected] LEGALNOTICE Thepublisherisnotresponsiblefortheusewhichmightbemadeofthefollowinginformation. PRINTEDINTHENETHERLANDS AndrzejMostowskiandFoundationalStudies v A.Ehrenfeucht,V.W.MarekandM.Srebrny(Eds.) IOSPress,2008 ©2008TheauthorsandIOSPress.Allrightsreserved. Preface AndrzejMostowskiwasoneoftheworld’sleadingEuropeanscientists,preservingand expanding a national school of research: in his case the Polish School of Logic. This effort took place within the adverse political and social circumstances of post-World War II Warsaw. Having taught at the underground Warsaw University during the War, Mostowski was active in rebuilding the teaching and research infrastructure in Poland, basedonthetraditionoflogicandmathematicsresearchthathadalreadyledtothecoun- try’s outstanding reputation before the War. The country’s world-wide contribution to research and teaching in logic, mathematics and computer science after the War bear witnesstotheimpactthatthisLogicSchool,ProfessorMostowski,hiscollaboratorsand studentshavehad.Apartfrombeingakeypersoninteachingandresearchmanagement, Mostowskiwasaprofoundresearcherwithcontributionsinthemainareasofmathemat- icallogicandthefoundationsofmathematics. This volume is a collection of fifteen research and expository articles, a complete bibliographyofAndrzejMostowski’swritings,threebiographicalandhistoricalarticles, andelevenshortpersonalreminiscences,allaimedatilluminatingAndrzejMostowski’s ideas and personality.Our mainmotivationfor solicitingand editingthese articles was thatthename“Mostowski”arousesagreatdealoffavorableinterestinverymanypeople fromvariouscountriesandinmanydifferentresearchcommunities. A starting point for anyone interested in the work of Mostowski is his famous se- riesoflectures“Thirtyyearsoffoundationalstudies”,publishedin1965,consideredby many as an ultimate presentation of the then-current state of mathematical logic, and moregenerally,thefoundationsofmathematics.MichaelDunn(logicianandcomputer scientist,IndianaUniversity,Bloomington)toldusthathecameacross“Thirtyyears...” asastudentinPittsburghandfoundtheworksointerestingandbeautifulthathecarried iteverywhereforthenexttwoyears,tryingtoconvinceeveryonearoundhimtoreadit. ItisnotclearwhetherMostowskiwouldbehappywiththetitleofthecurrentcol- lection. Modest as he was, he could very well argue that the scope of the foundations of mathematics is too large these days to be encompassed by a single volume and a singleresearcher.Theexplosionoffoundationalresearchresultingfromchallengesand progress in computer science and artificial intelligence makes it almost impossible to presenteventheirmainideasinasinglebook.Theirimpactisevenfeltfurtherafield,in economics,socialscience,engineering,cognitivesciencesandphilosophy. Thecontributionsincludedinthiscollectionpresentavarietyofapproachestodif- ferent areas of logic and the foundations of mathematics. Theyare written on different levels,byvariousresearcherspresentingavarietyofviews.Webelievethatthesearticles cancontinuetostimulateresearchintheareaspioneeredandmasteredbyMostowski. Reflecting Mostowski’s diverse interests, the research contributions in this volume relatetomanydistinctareas. Balcar and Jech study a class of Boolean algebras and its relationship with tech- niquesusedinsettheory,inparticularforcing. vi DickmannandPetrovichexhibitanimportantinteractionbetweenthree-valuedlogic and the algebraic theory of diagonal quadratic forms over a kind of rings, called semi- real. Relating to the original studies of Mostowski in the late 40’s and 50’s of the 20th century, Friedman discusses the current progress in Gödel phenomena, mainly incom- pleteness. Friedman lays out a program of significant progress that needs to be made in our comprehension of the scope of incompleteness. The paper challenges the foun- dationalcommunitywithspecificquestionsthatneedtobestudiedtoachieveadeeper understandingofincompleteness. GrzegorczykandZdanowskidevelopbasicdecidabilityresultswithoutresortingto Gödel’sencodingofformulasasintegers.Instead,theyworkwiththeelementarytheory ofstrings,introducedbyTarskiunderthenameof“concatenation”inhiscelebratedpaper ontheconceptoftruth.Theyprovethatthetheoryofstringsisessentiallyundecidable. Guzicki and Krynicki provide a new result in the area of generalized quantifiers. They define and pursue a quantifier that does not meet the very general definition of a quantifierproposedbyMostowskiin1957. KeislerextendscertainresultsofMostowskionthecomplexityclassoftheconcept of limz→∞F(z) = ∞, showing that in many structures the limit cannot be defined with fewer than three quantifier blocks, but in some more powerful structures the limit propertyforarbitraryfunctionscanbedefinedinbothtwo-quantifierforms. Knightpresentsthecurrentstateofresearchintheareasofthearithmetic(Kleene– Mostowski) and hyperarithmetic (Davis–Mostowski) hierarchies. In recent years, re- searchinthisareahasmergedwitheffectiveaspectsofmodeltheory,resultinginava- riety of deep results on the complexity of algebraic structures and constructions. The currentresearchinthisarea,sometimescalled“recursivemathematics”ispresented. Kotlarski deals with the interplay of syntactic and semantic arguments in Peano Arithmeticanditsfragments.Hispaperoutlinessignificantprogressinthestudyoffor- malizedarithmeticsinceMostowski’sdeath.Theresultsoftheseinvestigationshavere- visedourunderstandingofthenatureofindependentformulasinPeanoArithmetic,ex- hibiting sentences that shattered the conviction (current in some mathematical circles) that“ordinarycombinatorialsentences”cannotbeindependentfromPeanoArithmetic. ARamsey-likesentenceofthissortwasdiscoveredbyParisandHarrington.Numerous otherexampleswerefoundbyFriedmanandothers. Makowsky deals both with the reminiscences of Mostowski in the social con- text of doing mathematics, and with specific mathematical developments grounded in Mostowski’s work. Those developments, particularly in model theory and in abstract model theory, where Makowsky contributed significantly, have found application, out- sidepuremathematics,intheoreticalcomputerscience. Murawski and Wolen´ski investigate the philosophy of mathematics espoused in Mostowski’spapers.Mostowski’sworkwasfirmlygroundedinthephilosophicaltradi- tionofhisteachers,includingTarskiandKotarbin´ski.TheauthorsshowhowMostowski relatedtothefoundationalproblemsinmanyofhiscontributions,especiallyinhiswork andpresentationsofGödel’sincompletenessandCohen’sindependenceresults. Mycielski investigates the issue of mathematical truth, its formal and informal as- pects,anditsrelationshipwithmathematicalpractice.ThepaperdiscussesTarski’sthe- oryoftruthandthequestionofthepresenceofthehierarchyofmetatheoriesineveryday mathematics,metatheoriesdeterminedbysomeboundsontheallowedlengthsofproofs. vii L’InnocenteandMacintyrepresentaveryniceapplicationofamodel-theoreticresult ofMostowskiondirectproductsofmodelstocertainfundamentalproblemsofaspecial LiealgebraandtheDiophantinegeometryofcurves. The two expository papers by Scott and by Jankowski and Skowron are devoted to the algebraic approach to formal logic. Scott focuses on the quantifiers interpreted astheinfsandsups,respectively,inthealgebrasinterpretingvariouslogics.Jankowski and Skowron sketch the past and present perspective of the role of algebraic logic and Pawlak’sroughsetsinartificialintelligence. Wellsdiscussestheissueofpseudorecursivevarieties,thatis,varietiesV ofalgebras suchthatforeachnaturalnumbern,thesetofequationswithatmostnvariablestruein allalgebrasofV isrecursive,butsuchthatthesetofequationstrueinallalgebrasofV isnotrecursive.Thepaperoutlinesprogressintheareaaswellasanumberofimportant problems. Some of the contributors pointed out to us that their topics and results are derived from Alfred Tarski’s inspiration. We quickly agreedto underlinethe more general per- spective of the Tarski/Mostowski Berkeley/Warsaw school. The reader will find refer- ences to Tarski’s work, ideas and inspirations in several contributions in this volume. TheworkandvisionofAlfredTarski,albeitfromafaraftertheWWII,significantlyin- fluenced both Mostowski’s research and that of his collaborators. It is natural to treat Mostowski as “the Prince of the Tarskian kingdom”, as B.F. Wells, the last of Tarski’s students,doesinhiscontribution.(Inspiteofmanyeffortswehavenotbeenabletotrace any letter of the postal correspondence between Tarski and Mostowski. We do not be- lievethatthetwogreatmathematiciansjustdisposedoftheircorrespondence.Weleave itasanintriguingquestiontothehistoriansofscience.) The memory section of the volume consists of eleven very nice reminiscence arti- clesbyfamouslogicians(mathematicians,philosophersandcomputerscientists)Addi- son,Blikle,Dickmann,Feferman,Hajék,Kowalski,Wells,Vopeˇnkaandothers,shedding someinterestinglightontheirownresearchinterestsandachievements. Aswecontemplatethememoryofagiantoffoundationalresearch,themainques- tionthatpresentsitselfisthis: Howdidithappenthatoneman,locatedinarathersmallEuropeancountry,acoun- trywhoserulersdeliberatelyisolateditfromthemainstreamofworldscience,acountry ravagedbyasavagewar,wasabletocreateacenterforresearchthatextendeditsreach worldwide? TheanswertothisquestionliesinMostowski’spersonality,inhisabilitytofindhis way in the complex circumstances of local and world politics on the one hand, and on theotherhandinhisabilitytousetheopportunitiescreatedbythesamecircumstances toattractlargenumberofcollaborators,individualswhofounditappealingtoworkina largeresearchgroupunitedinitsgoalofbroadeningthehorizonsofhumanknowledge inlogicandthefoundationsofmathematics. The historic circumstances demanded such a man. The steady progress in the first half of the 20th centurycreated an opportunity– major problems that were unresolved relatedtotheverynatureofcomputationandreasoning(completenessandincomplete- nessphenomena,andthestudyofcomputablefunctions),tothefundamentalstructures underlying mathematics (sets, algorithms and epistemological principles of their exis- tence and manipulation), and to anticipation of the digital revolution that was about to happen. viii Theseopportunitieswereopentomany.ItwasMostowskiwho,consciouslyornot, transformed this potential into a research program for a large group of people who at- tached themselves locally, and who came as pilgrims from abroad, to take part in this uniqueexperience.ItisthroughMostowski’sdeepresearchresultsandthoseindividuals, theirstudentsandtheirscientificdescendantsthattheideasofMostowskilivetoday. Theeditorswouldliketothankallthecontributors.Wearesorrywecouldnotinvite moreexpertsduetothespacelimitations.Wewillgladlywelcomeanyformofcontin- uationofthiseffortreflectingAndrzejMostowski’simpact.Onesuchexample,entirely independent of the present volume, was a conference 50 Years of Generalized Quan- tifiers – in Honour of Professor Andrzej Mostowski, Warsaw, June 25 – July 1, 2007 (M.Krynicki,M.Mostowski,J.VäänänenandK.Zdanowski,organizers). This volume was conceived at a conference in Warsaw and Ruciane in September 2005.ThatmeetingcommemoratedthecontributionsofAndrzejMostowski(onthe30th anniversaryofhispassing),HelenaRasiowaandCecyliaRauszer.Severalcontributions collectedinthisvolume,includingD.Scott’sandV.Marek’s,werepresentedduringthe meeting. The editors express their thanks to the organizers of that meeting, including R.Wojcicki,A.SalwickiandG.Mirkowska-Salwicka. TheeditorsofthisvolumewereMostowski’sstudentsandcollaboratorsduringthe years(1954–1965AE),(1962–1975VWM)and(1969–1975MS). Wethankmanyindividualswhohelpedusintheprocessofeditingthisvolume,in- cludingtherefereeswhoprovidedmanyessentialcommentsandimprovedpresentations ofthecontributions. We extend our thanks to E. Fredriksson (like ourselves, a scientific descendant of Mostowski)andhisIOSPressofAmsterdam.Hislong-lastingencouragement,support andeffortsmadethisprojectpossibleandenabledthisvolumetocometofruition.Itis notanaccidentthatIOSpublishesmanytitlesthatrelatetothefoundationalresearch– seethecontributionofE.FredrikssonforadescriptionofhistiestoMostowski. Our very special thanks go to the IOS Press staff and M. Fröhlich, responsible for finaltypesettingandtheproductionofthevolume. September,2007 AndrzejEhrenfeucht,VictorMarek,MarianSrebrny ix Contents Preface AndrzejEhrenfeucht,VictorW.MarekandMarianSrebrny v Section1.History 1 OntheLifeandWorkofAndrzejMostowski(1913–1975) StanisławKrajewskiandMarianSrebrny 3 MathematicalLogicinWarsawinthe60’sand70’s,ortheInteractionofLogic andLife VictorW.Marek 15 MathematicalLogicinWarsaw:1918–1939 JanWolen´ski 30 Section2.MathematicalContributions 47 TheThree-ValuedLogicofQuadraticFormTheoryoverRealRings MaxDickmannandAlejandroPetrovich 49 RemarksonGödelPhenomenaandtheFieldofReals HarveyM.Friedman 68 UndecidabilityandConcatenation AndrzejGrzegorczykandKonradZdanowski 72 WitnessQuantifiersand0-1Laws WojciechGuzickiandMichałKrynicki 92 LogicforArtificialIntelligence:ARasiowa–PawlakSchoolPerspective AndrzejJankowskiandAndrzejSkowron 106 ContributionstotheTheoryofWeaklyDistributiveCompleteBooleanAlgebras BohuslavBalcarandThomasJech 144 QuantifiersinLimits H.JeromeKeisler 151 TheKleene–MostowskiHierarchyandtheDavis–MostowskiHierarchy JuliaF.Knight 171 AModelTheoreticApproachtoProofTheoryofArithmetic HenrykKotlarski 198 TowardsDecidabilityoftheTheoryofPseudo-FiniteDimensional Representationsofsl (k);I 2 SoniaL’InnocenteandAngusMacIntyre 235 x EncounterswithA.Mostowski JohannA.Makowsky 261 ObservationsonTruth,ConsistencyandLengthsofProofs JanMycielski 278 TheAlgebraicInterpretationofQuantifiers:IntuitionisticandClassical DanaS.Scott 289 AbstractingandGeneralizingPseudorecursiveness BenjaminWells 313 AndrzejMostowskiontheFoundationsandPhilosophyofMathematics RomanMurawskiandJanWolen´ski 324 Section3.BibliographyofAndrzejMostowski 339 BibliographyofAndrzejMostowski JanZygmunt 341 Section4.Memories 373 Warsaw1957:MemoriesofMostowski JohnW.Addison 375 AndrzejMostowski–MyMasterinMathematics AndrzejJacekBlikle 380 OurReminiscencesofAndrzejMostowski EmiliodelSolarandMaxDickmann 384 AndrzejMostowski:AnAppreciation SolomonFeferman 388 AndrzejMostowskiasTeacherandEditor EinarH.Fredriksson 393 MostowskiandCzech–PolishCooperationinMathematicalLogic PetrHájek,VictorW.MarekandPetrVopeˇnka 403 ReminiscencesofWarsawandLogic,1964 PeterG.Hinman 406 FromMathematicalLogic,toNaturalLanguage,ArtificialIntelligence,and HumanThinking(AShortEssayinHonourofAndrzejMostowski) RobertKowalski 410 AVeryPersonalRecollection HidemitsuSayeki 414 ThePrinceofLogic BenjaminWells 421
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