Model Theory : An Introduction

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Springer Science & Business Media, Aug 21, 2002 - Mathematics - 342 pages
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Model theory is a branch of mathematical logic where we study mathem- ical structures by considering the ?rst-order sentences true in those str- turesandthesetsde?nableby?rst-orderformulas.Traditionallytherehave been two principal themes in the subject: •startingwithaconcretemathematicalstructure,suchasthe?eldofreal numbers, and using model-theoretic techniques to obtain new information about the structure and the sets de?nable in the structure; • looking at theories that have some interesting property and proving general structure theorems about their models. A good example of the ?rst theme is Tarski’s work on the ?eld of real numbers. Tarski showed that the theory of the real ?eld is decidable. This is a sharp contrast to G ̈ odel’s Incompleteness Theorem, which showed that the theory of the seemingly simpler ring of integers is undecidable. For his proof, Tarski developed the method of quanti?er elimination which can be n used to show that all subsets of R de?nable in the real ?eld are geom- rically well-behaved. More recently, Wilkie [103] extended these ideas to prove that sets de?nable in the real exponential ?eld are also well-behaved. ThesecondthemeisillustratedbyMorley’sCategoricityTheorem,which says that if T is a theory in a countable language and there is an uncou- able cardinal ? such that, up to isomorphism, T has a unique model of cardinality ?,then T has a unique model of cardinality ? for every - countable?.ThislinehasbeenextendedbyShelah[92],whohasdeveloped deep general classi?cation results.
  

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A great introduction for anyone interested in studying applied model theory. Applications to algebra, in particular model theory of fields, are especially well- presented. The exercises are interesting and plentiful.
The book is perfect preparation for studying more advanced topics in applied model theory. The book also is good preparation for studying stability theory in more general cases (in this book, the totally transcendental or omega stable (the languages here are countable here) cases are carefully presented). I think studying this book leads naturally to studying stability theory, more advanced model theory of fields, and other applied model theory (like say o-minimality). One of the best aspects of the text is the historical section at the end of each chapter. These sections often mention current avenues of research in addition to describing the history of the work in the chapter.
For other model theory topics, like model theory of modules or groups, there may be more suitable texts like Poizat's book (though one should be warned that the language of model theory has evolved, and the modern working language is used in this text, but not in older ones). For those not interested in becoming model theorists, but interested in picking up some interesting model theory and applications to their own branch of mathematics, this is the ideal book. Marker's text is much better suited to this sort of study than, say, Chang and Keisler's text or the more modern book by Hodges.
Additionally, it should be said that Dave Marker's writing style is very nice, keeping the reader excited and organized. Professor Marker won the 2007 Inaugural Shoenfield Prize from the ASL for this textbook.
 

Contents

II
7
III
14
IV
19
V
29
VI
33
VII
40
VIII
44
IX
48
XXIV
202
XXV
207
XXVI
215
XXVII
227
XXVIII
236
XXIX
240
XXX
243
XXXI
251

X
60
XI
71
XII
84
XIII
93
XIV
104
XV
115
XVI
125
XVII
138
XVIII
155
XIX
163
XX
175
XXI
178
XXII
189
XXIII
195
XXXII
255
XXXIII
261
XXXIV
267
XXXV
279
XXXVI
285
XXXVII
289
XXXVIII
293
XXXIX
300
XL
309
XLI
315
XLII
323
XLIII
329
XLIV
337
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About the author (2002)

David Marker is Professor of Mathematics at the University of Illinois at Chicago.

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