Graduate Texts in Mathematics Ser.: Model Theory : An Introduction by David Marker (2002, Hardcover)

Springer New York

0387987606

9780387987606

2274201

VIII, 345 Pages

English

Model Theory : an Introduction

2002

Logic

Textbook

David Marker

Mathematics

Graduate Texts in Mathematics Ser.

Hardcover

24.7 Oz

9.3 in

6.1 in

Scholarly & Professional

2002-024184

23

From the reviews:MATHEMATICAL REVIEWS"This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics…There is a strong focus on the meaning of model-theoretic concepts in mathematically interesting examples. The exercises touch on a wealth of beautiful topics.""This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics, with a route leading to a substantial treatment of Hrushovski's proof of the Mordell-Lang conjecture for function fields. … The exercises touch on a wealth of beautiful topics. … There is additional basic background in two appendices (on set theory and on real algebra)." (Dugald Macpherson, Mathematical Reviews, 2003 e)"Model theory is the branch of mathematical logic that examines what it means for a first-order sentence … to be true in a particular structure … . This is a text for graduate students, mainly aimed at those specializing in logic, but also of interest for mathematicians outside logic who want to know what model theory can offer them in their own disciplines. … it is one which makes a good case for model theory as much more than a tool for specialist logicians." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (513), 2004)"The author's intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications. … The text is noteworthy for its wealth of examples and its desire to bring the student to the point where the frontiers of research are visible. … this book should be on the shelf of anybody with an interest in model theory." (J. M. Plotkin, Zentralblatt Math, Vol. 1003 (03), 2003), From the reviews: MATHEMATICAL REVIEWS "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics...There is a strong focus on the meaning of model-theoretic concepts in mathematically interesting examples. The exercises touch on a wealth of beautiful topics." "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics, with a route leading to a substantial treatment of Hrushovski's proof of the Mordell-Lang conjecture for function fields. ... The exercises touch on a wealth of beautiful topics. ... There is additional basic background in two appendices (on set theory and on real algebra)." (Dugald Macpherson, Mathematical Reviews, 2003 e) "Model theory is the branch of mathematical logic that examines what it means for a first-order sentence ... to be true in a particular structure ... . This is a text for graduate students, mainly aimed at those specializing in logic, but also of interest for mathematicians outside logic who want to know what model theory can offer them in their own disciplines. ... it is one which makes a good case for model theory as much more than a tool for specialist logicians." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (513), 2004) "The author's intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications. ... The text is noteworthy for its wealth of examples and its desire to bring the student to the point where the frontiers of research are visible. ... this book should be on the shelf of anybody with an interest in model theory." (J. M. Plotkin, Zentralblatt Math, Vol. 1003 (03), 2003), From the reviews: MATHEMATICAL REVIEWS "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics…There is a strong focus on the meaning of model-theoretic concepts in mathematically interesting examples. The exercises touch on a wealth of beautiful topics." "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics, with a route leading to a substantial treatment of Hrushovski's proof of the Mordell-Lang conjecture for function fields. … The exercises touch on a wealth of beautiful topics. … There is additional basic background in two appendices (on set theory and on real algebra)." (Dugald Macpherson, Mathematical Reviews, 2003 e) "Model theory is the branch of mathematical logic that examines what it means for a first-order sentence … to be true in a particular structure … . This is a text for graduate students, mainly aimed at those specializing in logic, but also of interest for mathematicians outside logic who want to know what model theory can offer them in their own disciplines. … it is one which makes a good case for model theory as much more than a tool for specialist logicians." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (513), 2004) "The author's intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications. … The text is noteworthy for its wealth of examples and its desire to bring the student to the point where the frontiers of research are visible. … this book should be on the shelf of anybody with an interest in model theory." (J. M. Plotkin, Zentralblatt Math, Vol. 1003 (03), 2003)

217

1 vol.

Yes

511.3

Structures and Theories.- Basic Techniques.- Algebraic Examples.- Realizing and Omitting Types.- Indiscernibles.- ?-Stable Theories.- ?-Stable Groups.- Geometry of Strongly Minimal Sets.

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., This book offers an introductory presentation on model theory emphasizing connections to algebra. It will be an appropriate introduction both for graduate students interested in advanced work in model theory and for students and researchers in logic or algebra who want to learn the basic results and themes of model theory. In the end, the reader will have a firm background in model theory and be well motivated and well prepared for more advanced treatments like Pillay's Geometric Model Theory or Buechler's Essential Stability Theory., From the reviews: "This is an extremely fine graduate-level textbook on model theory. There is a careful selection of topics. There is a strong focus on the meaning of model-theoretic concepts in mathematically interesting examples. The exercises touch on a wealth of beautiful topics." Mathematical Reviews, Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures, Preliminary Text. Do not use. This book offers an introductory course in model theory emphasizing connections to algebra. It will be an appropriate introduction both for graduate students interested in advanced work in model theory and for students and researchers in logic or algebra who want to learn the basic results and themes of model theory. In the end, the reader will have a firm background in model theory and be well motivated and well prepared for more advanced treatments like Pillay's "Geometric Model Theory" or Buechler's "Essential Stability Theory."

QA8.9-10.3