Mathematical Logic

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Springer, Jun 10, 1994 - Mathematics - 308 pages
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This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. FraissÚ's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic.

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About the author (1994)

Prof. Jvrg Flum, Abteilung f]r Mathematische Logik, Albert-Ludwigs-Universitdt Freiburg, Germany, http: // Prof. Martin Grohe, Institut f]r Informatik, Humboldt-Universitdt zu Berlin, Germany, http: // The authors are very well qualified to write this book. In addition to their strong backgrounds in complexity, algorithms, etc., they have contributed a number of specific key results in parameterized complexity (e.g., http: // Jvrg Flum has coauthored two other Springer monographs: (i) "Mathematical Logic," Undergraduate Texts in Mathematics, 0-387-94258-0, 3rd printing since 1994, over 4000 copies sold, Heinz-Dieter Ebbinghaus, Jvrg Flum, Wolfgang Thomas, http: // (ii) "Finite Model Theory," Springer Monographs in Mathematics (was in series Perspectives in Mathematical Logic), printed in soft- and hardback, 1995, 2nd ed. in 1999, 2nd corr. print in 2006, Heinz-Dieter Ebbinghaus, Jvrg Flum, 3-540-28787-6, http: // In addition, Jvrg Flum coauthored the following LNM title: Vol. 769, "Topological Model Theory, 1980, 3-540-09732-5, Jvrg Flum, Martin Ziegler. And he coedited the following LNCS title: Vol. 1683, CSL 1999 conf. proc., Jvrg Flum, Mario Rodriguez-Artalejo, 1999, 3-540-66536-6. Prof. Martin Grohe has authored over 50 articles for refereed theoretical computer science journals and conference proceedings (http: // Martin.html) in the areas of logic, complexity, algorithms, etc.

Prof. Dr. Heinz-Dieter Ebbinghaus und Prof. Dr. JArg Flum forschen und lehren am Institut fA1/4r Mathematik der UniversitAt Freiburg, Prof. Dr. Wolfgang Thomas ist Inhaber des Lehrstuhls fA1/4r Informatik 7 (Logik und Theorie diskreter Systeme) der RWTH Aachen.

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