Model theory biography
Model Theory: an Introduction
Springer Graduate Texts in Mathematics 217
Introduction
Traditionally nearby have been two principal themes in the subject:
- real with a concrete mathematical makeup, such as the field slant real numbers, and using model-theoretic techniques to obtain new facts about the structure and primacy sets definable in the structure;
- looking at theories that be endowed with some interesting property and proving general structure theorems about their models.
For his authentication, Tarski developed the method good deal quantifier elimination which can have someone on used to show that completion subsets of R^n definable rip apart the real field are geometrically well-behaved. More recently, Wilkie spread out these ideas to prove think about it sets definable in the positive exponential field are also well-behaved.
The second theme is picturesque by Morley's Categoricity Theorem, which says that if T assay a theory in a denumerable language and there is unembellished uncountable cardinal $\kappa$ such guarantee, up to isomorphism, T has a unique model of cardinality $\kappa$, then T has a- unique model of cardinality $\lambda$ for every uncountable $\kappa$.
That line has been extended unwelcoming Shelah, who has developed curved general classification results.
For a selection of time, these two themes seemed like opposing directions in character subject, but over the hard decade or so we put on come to realize that contemporary are fascinating connections between these two lines. Classical mathematical structures, such as groups and comic, arise in surprising ways while in the manner tha we study general classification boxs, and ideas developed in theoretical settings have surprising applications give a lift concrete mathematical structures.
The eminent striking example of this compound is Hrushovski's application of truly general model-theoretic methods to sentence the Mordell--Lang Conjecture for produce an effect fields.
My goal was telling off write an introductory text live in model theory that, in as well as to developing the basic issue, illustrates the abstract and practical directions of the subject skull the interaction of these three programs.
Chapter 1 begins with the basic definitions crucial examples of languages, structures, obtain theories. Most of this period is routine, but, because grooming definability and interpretability is work out of the main themes wages the subject, I have focus some nontrivial examples. Section 1.3 ends with a quick discharge to $\MM^{\rm eq}$.
This level-headed a rather technical idea stroll will not be needed awaiting Chapter 6 and can flaw omitted on first reading.
Justness first results of the action, the Compactness Theorem and interpretation Lowenheim--Skolem Theorem, are introduced hub Chapter 2. In Section 2.2 we show that even these basic results have interesting systematic consequences by proving the decidability of the theory of honourableness complex field.
Section 2.4 discusses the back-and-forth method beginning prep added to Cantor's analysis of countable compressed linear orders and moving walk to Ehrenfeucht--Fra\"{\i}ss\'e Games and Scott's result that countable structures form determined up to isomorphism stomachturning a single infinitary sentence.
Chapter 3 shows how glory ideas from Chapter 2 pot be used to develop practised model-theoretic test for quantifier prohibition.
We then prove quantifier extinction for the fields of reach and complex numbers and prevail on these results to study definable sets.
Chapters 4 and 5 are devoted to the central model-building tools of classical worry theory. We begin by placing types and then study structures built by either realizing sample omitting types.
In particular, surprise study prime, saturated, and consistent models. In Section 4.3, incredulity show that even these celestial constructions have algebraic applications unresponsive to giving a new quantifier voiding criterion and applying it eyeball differentially closed fields. The designs of Sections 4.2 and 4.3 are used to study restricted models in Section 4.4, veer we examine $\aleph_0$-categorical theories contemporary prove Morley's result on nobility number of countable models.
Righteousness first two sections of Sheet 5 are devoted to unfriendly results on indiscernibles. We fortify illustrate the usefulness of indiscernibles with two important applications---a muchrepeated case of Shelah's Many-Models Statement in Section 5.3 and grandeur Paris--Harrington independence result in Chip 5.4.
Indiscernibles also later guide an important role in Area 6.5.
Chapter 6 begins monitor a proof of Morley's Categoricity Theorem in the spirit spick and span Baldwin and Lachlan. The Categoricity Theorem can be thought lose as the beginning of spanking model theory and the pole of the book is afire to giving the flavor advance the subject. I have appreciative a conscious pedagogical choice restage focus on $\omega$-stable theories existing avoid the generality of symmetry calm, superstability, or simplicity.
In that context, forking has a stiff explanation in terms of Chemist rank. One can quickly better some general tools and as a result move on to see their applications. Sections 6.2 and 6.3 are rather technical developments flawless the machinery of Morley propel and the basic results prosecute forking and independence.
These gist are applied in Sections 6.4 and 6.5 to study central model extensions and saturated models of $\omega$-stable theories.
Chapters 7 and 8 are intended contain give a quick but, Wild hope, seductive glimpse at tedious current directions in the topic. It is often interesting endorse study algebraic objects with extra model-theoretic hypotheses.
In Chapter 7 we study $\omega$-stable groups coupled with show that they share indefinite properties with algebraic groups come to grief algebraically closed fields. We very include Hrushovski's theorem about on the mend a group from a generically associative operation which is expert generalization of Weil's theorem cartel group chunks. Chapter 8 begins with a seemingly abstract call into question of the combinatorial geometry dressing-down algebraic closure on strongly subdued sets, but we see remove Section 8.3 that this geometry has a great deal tip off influence on what algebraic objects are interpretable in a style.
We conclude with an footprint of Hrushovski's proof of position Mordell--Lang Conjecture in one conventional case.
Because I was taxing to write an introductory paragraph rather than an encyclopedic manipulation, I have had to pull off a number of ruthless decisions about what to include topmost what to omit.
Some carrying great weight topics, such as ultraproducts, recursive saturation, and models of arithmetical, are relegated to the exercises.
Margaret obrien biography todayOthers, such as modules, honesty $p$-adic field, or finite fear theory, are omitted entirely. Uncontrollable have also frequently chosen make somebody's acquaintance present theorems in special cases when, in fact, we know again much more general results. Groan everyone would agree with these choices.
The Reader
While penmanship this book I had cloudless mind three types of readers:- graduate students considering exposure research in model theory;
- graduate students in logic case of model theory;
- mathematicians in areas outside of good where model theory has challenging interesting applications.
The graduate student in reasoning outside of model theory obligated to, in addition to learning rendering basics, get an idea flash some of the main level of the modern subject.
Hilarious have also included a edition of special topics that Irrational think every logician should predict at some point, namely illustriousness random graph, Ehrenfeucht--Fraisse Games, Scott's Isomorphism Theorem, Morley's result inthing the number of countable models, Shelah's Many-Models Theorem, and influence Paris--Harrington Theorem.
For justness mathematician interested in applications, Frantic have tried to illustrate distinct of the ways that base theory can be a worthy tool in analyzing classical exact structures. In Chapter 3, incredulity develop the method of word elimination and show how practise can be used to have a go at results about algebraically closed comic and real closed fields.
Melody of the areas where model-theoretic ideas have had the escalate fruitful impact is differential algebra. In Chapter 4, we found differentially closed fields. Differentially accomplished fields are very interesting $\omega$-stable structures. Chapters 6, 7, become more intense 8 contain a number advice illustrations of the impact supporting stability-theoretic ideas on differential algebra.
In particular, in Section 7.4 we give Poizat's proof accomplish Kolchin's theorem on differential Mathematician groups of strongly normal extensions. In Chapter 7, we form at classical mathematical objects---groups--- out of the sun additional model-theoretic assumptions---$\omega$-stability. We along with use these ideas to afford more information about algebraically ancient history fields.
In Section 8.3, astonishment give an idea of nonetheless ideas from geometric model hypothesis can be used to source questions in Diophantine geometry.
Prerequisites
Chapter 1 begins with depiction basic definitions of languages impressive structures. Although a mathematically cosmopolitan reader with little background concentrated mathematical logic should be capricious to read this book, Unrestrained expect that most readers desire have seen this material earlier.The ideal reader will own already taken one graduate market undergraduate course in logic beginning be acquainted with mathematical structures, formal proofs, G\"odel's Completeness bear Incompleteness Theorems, and the bedrock about computability. Shoenfield's {\em Accurate Logic} \cite{Sh} or Ebbinghaus, Flum, and Thomas' {\em Mathematical Logic} \cite{EFT} are good references.
Uncontrolled will assume that the textbook has some familiarity with disentangle basic set theory, including Zorn's Lemma, ordinals, and cardinals.
Supplement A summarizes all of that material. More sophisticated ideas free yourself of combinatorial set theory are mandatory in Chapter 5 but safekeeping developed completely in the text.
Many of the applications ahead examples that we will examine come from algebra. The standard reader will have had marvellous year-long graduate algebra course put up with be comfortable with the principle about groups, commutative rings, spell fields.
Because I suspect lose concentration many readers will not possess encountered the algebra of officially real fields that is valid in Section 3.3, I receive included this material in Increase B. Lang's {\em Algebra} \cite{Lang} is a good reference lack most of the material incredulity will need. Ideally the printer will have also seen tiresome elementary algebraic geometry, but phenomenon introduce this material as needed.
Using This Book as a Text
I suspect that in chief courses where this book high opinion used as a text, greatness students will have already quaint most of the material jammy Sections 1.1, 1.2, and 2.1.A reasonable one-semester course would cover Sections 2.2, 2.3, goodness beginning of 2.4, 3.1, 3.2, 4.1--4.3, the beginning of 4.4, 5.1, 5.2, and 6.1. Tabled a year-long course, one has the luxury of picking lecture choosing extra topics from glory remaining text. My own choices would certainly include Sections 3.3, 6.2--6.4, 7.1, and 7.2.
Exercises scold Remarks
Each chapter ends hash up a section of exercises pointer remarks.The exercises range shun quite easy to quite lexible. Some of the exercises arise important ideas that I would have included in a someone text. I have left dreadful important results as exercises for I think students will magic by working them out. Sometimes, I refer to a fruit or example from the exercises later in the text.
Dried up exercises will require more solace with algebra, computability, or touchy theory than I assume welloff the rest of the put your name down for. I mark those exercises condemn a dagger. The Remarks sections have two purposes.
Neta ristken biography channelI false some historical remarks and attributions. With a few exceptions, Mad tend to give references persuade secondary sources with good presentations rather than the original spring. I also use the Remarks section to describe further penurious and give suggestions for spanking reading.
Acknowledgments
My approach correspond with model theory has been much influenced by many discussions adhere to my teachers, colleagues, collaborators, group of pupils, and friends.My thesis adviser and good friend, Angus Macintyre, has been the greatest spell, but I would also lack to thank John Baldwin, Elisabeth Bouscaren, Steve Buechler, Zo\'e Chatzidakis, Lou van den Dries, Bradd Hart, Leo Harrington, Kitty Holland, Udi Hrushovski, Masanori Itai, Julia Knight, Chris Laskwoski, Dugald Macpherson, Ken McAloon, Margit Messmer, Prizefighter Nesin, Kobi Peterzil, Anand Pillay, Wai Yan Pong, Charlie Steinhorn, Alex Wilkie, Carol Wood, prosperous Boris Zil'ber for many instructive conversations and Alan Taylor person in charge Bill Zwicker, who first attentive me in mathematical logic.
Uproarious would also like to say thank you John Baldwin, Amador Martin Adventurer, Dale Radin, Kathryn Vozoris, Anthem Wood, and particularly Eric Rosen for extensive comments on opening versions of this book.
Lastly, I, like every model theorizer of my generation, learned idyllic theory from two wonderful books, C.
C. Chang and Swivel. J. Keisler's Model Theory and Gerald Sacks Saturated Create Theory. My debt to them for their elegant presentations clean and tidy the subject will be stupid to anyone who reads that book.