Andrzej Mostowski and foundational studies

460 Pages·2008·2.14 MB·English

by  A. Ehrenfeucht

ANDRZEJ MOSTOWSKI AND
FOUNDATIONAL STUDIES
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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.
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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