Mathematical Logic

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Springer New York, Dec 1, 1996 - Mathematics - 291 pages
What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathe matical proofs? Only in this century has there been success in obtaining substantial and satisfactory answers. The present book contains a systematic discussion of these results. The investigations are centered around first-order logic. Our first goal is Godel's completeness theorem, which shows that the con sequence relation coincides with formal provability: By means of a calcu lus consisting of simple formal inference rules, one can obtain all conse quences of a given axiom system (and in particular, imitate all mathemat ical proofs). A short digression into model theory will help us to analyze the expres sive power of the first-order language, and it will turn out that there are certain deficiencies. For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms. We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner.

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About the author (1996)

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.