Rudiments of Calculus

Author: A. Arnold
Publisher: Elsevier
ISBN: 9780080516455
Format: PDF, Mobi
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This book presents what in our opinion constitutes the basis of the theory of the mu-calculus, considered as an algebraic system rather than a logic. We have wished to present the subject in a unified way, and in a form as general as possible. Therefore, our emphasis is on the generality of the fixed-point notation, and on the connections between mu-calculus, games, and automata, which we also explain in an algebraic way. This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially for researchers and students interested in logic in computer science, comuter aided verification, and general aspects of automata theory. We have aimed at gathering in a single place the fundamental results of the theory, that are currently very scattered in the literature, and often hardly accessible for interested readers. The presentation is self-contained, except for the proof of the Mc-Naughton's Determinization Theorem (see, e.g., [97]. However, we suppose that the reader is already familiar with some basic automata theory and universal algebra. The references, credits, and suggestions for further reading are given at the end of each chapter.

The Logical Foundations of Mathematics

Author: William S. Hatcher
Publisher: Elsevier
ISBN: 1483189635
Format: PDF, Mobi
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The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

Discrete Mathematics

Author: R. C. Penner
Publisher: World Scientific
ISBN: 9789810240882
Format: PDF, ePub, Docs
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This book offers an introduction to mathematical proofs and to the fundamentals of modern mathematics. No real prerequisites are needed other than a suitable level of mathematical maturity. The text is divided into two parts, the first of which constitutes the core of a one-semester course covering proofs, predicate calculus, set theory, elementary number theory, relations, and functions, and the second of which applies this material to a more advanced study of selected topics in pure mathematics, applied mathematics, and computer science, specifically cardinality, combinatorics, finite-state automata, and graphs. In both parts, deeper and more interesting material is treated in optional sections, and the text has been kept flexible by allowing many different possible courses or emphases based upon different paths through the volume.

G del 96

Author: Petr Hájek
Publisher: Cambridge University Press
ISBN: 1316739309
Format: PDF, Mobi
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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the sixth publication in the Lecture Notes in Logic series, collects the proceedings of the conference 'Logical Foundations of Mathematics, Computer Science, and Physics - Kurt Gödel's Legacy', held in Brno, Czech Republic, on the 90th anniversary of Gödel's birth. The broad range of speakers who participated in this event affirms the continuing importance of Gödel's work in logic, physics, and the philosophy and foundations of mathematics and computer science. The papers in this volume range over all these topics and contribute to our present understanding of them.