Model Theory in Algebra, Analysis and Arithmetic: Cetraro, Italy 2012, Editors: H. Dugald Macpherson, Carlo Toffalori

201 Pages·2014·1.93 MB·English

by  Lou van den Dries

Lecture Notes in Mathematics 2111
CIME Foundation Subseries
Lou van den Dries · Jochen Koenigsmann
H. Dugald Macpherson · Anand Pillay
Carlo Toffalori · Alex J. Wilkie
Model Theory in
Algebra, Analysis
and Arithmetic
Cetraro, Italy 2012
Editors: H. Dugald Macpherson,
Carlo Toffalori
Lecture Notes in Mathematics 2111
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
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FondazioneC.I.M.E.,Firenze
C.I.M.E.stands forCentroInternazionale Matematico Estivo,thatis,International Mathe-
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FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
Lou van den Dries (cid:129) Jochen Koenigsmann (cid:129)
H. Dugald Macpherson (cid:129) Anand Pillay (cid:129)
Carlo Toffalori (cid:129) Alex J. Wilkie
Model Theory in Algebra,
Analysis and Arithmetic
Cetraro, Italy 2012
Editors: H. Dugald Macpherson,
Carlo Toffalori
123
LouvandenDries JochenKoenigsmann
DepartmentofMathematics MathematicalInstitute
UniversityofIllinois UniversityofOxford
Urbana,IL,USA Oxford,UnitedKingdom
H.DugaldMacpherson AnandPillay
UniversityofLeedsSchoolofMathematics DepartmentofMathematics
Leeds,UnitedKingdom UniversityofNotreDame
Hurley,IN,USA
CarloToffalori AlexJ.Wilkie
DivisionofMathematics UniversityofManchesterSchool
UniversityofCamerinoSchoolofScience ofMathematics
andTechnology Manchester,UnitedKingdom
Camerino,Italy
ISBN978-3-642-54935-9 ISBN978-3-642-54936-6(eBook)
DOI10.1007/978-3-642-54936-6
SpringerHeidelbergNewYorkDordrechtLondon
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LibraryofCongressControlNumber:2014945978
MathematicsSubjectClassification(2010):03CXX;03C45;03C10;03C60;03C64;12J20;11U05;
11U09
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Acknowledgements
CIME activity is carried out with the collaboration and financial support of
the following Italian institutions: INdAM (Istituto Nazionale di Alta Matema-
tica, National Institute of High Mathematics), MIUR (Ministero dell’Istruzione,
dell’Universitàe della Ricerca,Ministeryof Education,UniversityandResearch),
andEnteCassa diRisparmiodiFirenze.ThisCIMECourseisalsosponsoredand
financiallysupportedbytheItalianresearchprojectFIR“Newtrendsinthemodel
theoryofexponentiation”.
v

Contents
ModelTheoryinAlgebra,AnalysisandArithmetic:APreface............ 1
DugaldMacphersonandCarloToffalori
Some Themes Around First Order Theories Without
theIndependenceProperty ..................................................... 13
AnandPillay
LecturesontheModelTheoryofRealandComplexExponentiation ..... 35
A.J.Wilkie
LecturesontheModelTheoryofValuedFields.............................. 55
LouvandenDries
UndecidabilityinNumberTheory ............................................. 159
JochenKoenigsmann
vii
Model Theory in Algebra, Analysis
and Arithmetic: A Preface
DugaldMacphersonandCarloToffalori
1 Meeting Model Theory
Model theory is a branch of mathematical logic dealing with mathematical struc-
tures (models) from the point of view of first order logical definability. Although
comparatively young, it is now well established, its major textbooks including
[6,17,34,43,53]. A typical goal of model theory is to build, study and classify
mathematical universes in which some given axioms (usually expressed in a first
order way) are satisfied. Thus model theory has remote roots in the birth of non-
Euclidean geometries and in the effort to realize them in suitable mathematical
settings where their revolutionary (non-standard) assumptions are obeyed. Some
majorachievementsofmathematicallogicinthefirsthalfofthetwentiethcentury,
such as the Gödel Compactness Theorem around 1930, underpin modern model
theory.
Itseemsthatthename“modeltheory”,moreprecisely“theoryofmodels”,was
explicitlyusedforthefirsttimebyAlfredTarskiin1954inthepapers[51,52].In
the 1950s, thanks to the work of Tarski himself, Abraham Robinson, Vaught, and
others,modeltheorystartedtoemergeasanewindependenttopicinmathematical
logic.
Today,modeltheoryhasasophisticatedandcurrentlyevolvinginternalabstract
theory.Ithasalsodeepeningandstrengtheninginteractionswith,forexample,parts
of algebra,combinatorics,geometry,numbertheoryand analysis, oftenleadingto
striking applications—this relationship with other branches of mathematics dates
D.Macpherson((cid:2))
SchoolofMathematics,UniversityofLeeds,LeedsLS29JT,UK
e-mail:[email protected]
C.Toffalori
SchoolofScienceandTechnology,DivisionofMathematics,UniversityofCamerino,62032
Camerino,Italy
e-mail:[email protected]
L.vandenDriesetal.,ModelTheoryinAlgebra,AnalysisandArithmetic,LectureNotes 1
inMathematics2111,DOI10.1007/978-3-642-54936-6__1,
©Springer-VerlagBerlinHeidelberg2014
2 D.MacphersonandC.Toffalori
backtoTarskihimselfandtheverybeginningofmodeltheory.Thesetwofeatures
areoftenlabelled‘puremodeltheory’and‘appliedmodeltheory’.Inthe1970sand
early 1980s they were sometimes pursued rather independently, but the pure and
appliedaspectsarenowcloselyintertwined,feedingoffandstimulatingeachother.
Central parts of the internal theory (e.g. types, saturation and homogeneity,
ultraproducts,categoricity,along with parts of infinitary logic) arose in the 1950s
and 1960s. The subject developed its modern flavour partly through the work of
Morley[38] andthenShelah[48]andothers,beguninthe1960s,whichlooksfor
abstract notions of classification, independence and dimension. Crucial concepts
were singled out, such as stability, simplicity, o-minimality (see [42,55,56], for
instance)andtheNIPproperty,andotherssuchastheNTP2propertyandgeneric
stabilityarejustbeginningtobedeveloped.Stabilitytheoryisatthecore.
Interactionsofmodeltheorywithotherpartsofmathematicscanbetracedback
to Tarski’s quantifier elimination for the complex and real field in the 1930s (see
[54]).Connectionsaredescribedinmanysources,suchas[7,8,15,31,32].Among
applications of model theory, we mention a few examples: o-minimality (and its
close connection with real analysis and real analytic and algebraic geometry);
Hrushovski’s work in Diophantine geometry [19], which yields new approaches
to the Mordell-Lang and Manin-Mumford Conjectures with new results [5]; the
recent work of Pila and Wilkie [41] on rational heights, together with further
work of Pila [40] and others, which makes striking links between o-minimality
and diophantinegeometry;motivic integrationas developedby Denef and Loeser
and subsequently Cluckers, Hrushovski, and Kazhdan (see [29] for a general
introductionandextendedbibliographicalreferences);andtherich ideasof Zilber
aroundthelogicofcomplexexponentiation[59]andZariskigeometries[60].
The present volume consists of four survey articles on contemporary themes
in model theory. It is based on notes from lecture courses by the same authors
at the meeting Model Theory in Algebra, Analysis and Arithmetic organized by
CIME (CentroInternazionaleMatematicoEstivo),July 2–6,2012,directedbythe
editors of the volume. The meeting continued the spirit of a much earlier CIME
courseModelTheoryandApplicationsin1975.ThelatterwascoordinatedbyPiero
Mangani,1withspeakersH.J.Keisler,G.Sabbagh,G.E.Sacks,J.A.Makowski,M.
Servi, C. Wood and A. Macintyre. Earlier, in 1968, a CIME course directed by
EttoreCasaridealtwithAspectsofMathematicalLogicandinparticularwithmodel
theory,itsspeakersincludingA.MostowskiandA.Robinsonhimself.
The CIME 2012 course was intended particularly for young researchers (e.g.
Ph.D.studentsandpostdocs),bothmodeltheorists,andthosefromoutsideintrigued
by recent applications and looking for an introduction to current model theory. It
alsohadaverywelcomeparticipationofmoreseniorresearchersinotherbranches
of mathematics, which encouraged cross-fertilization and was very beneficial for
younger researchers. The course complemented other recent activities in Europe
1PieroMangani diedonApril4,2013; thesecond author wouldliketodevotethiswork tohis
memory.