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Books > Science & Mathematics > Mathematics > Mathematical foundations > Mathematical logic
This new book by a renowned logician provides an introduction to self-reference and diagonalization, and presents a unified treatment of fixed points as they occur in Godels incompleteness proofs, recursion theory, combinatory logice, semantics, and metamethamatics. A survey of introductory material, metamathematics, and a summary of recent research are provided. A large number of exercises (with their solutions), are also provided in the introductory chapters.
Koennen Computer alles? Wenn es so ware, gabe es dieses Buch nicht. Es beweist bestechend logisch, dass selbst die groessten, schnellsten, intelligentesten und teuersten Computer der Welt nur beschrankt leistungsfahig sind. Der Mensch kann noch so viel Geld, Zeit und Know-how investieren, es gibt Computer-Probleme, die er niemals loesen wird. Eine beunruhigende, provokative Botschaft - und doch: wussten wir es nicht eigentlich schon, haben es aber nie wirklich glauben wollen? Der bekannte Computer-Wissenschaftler David Harel vermittelt die mathematischen Fakten spannend, unterhaltsam und allgemeinverstandlich. Mit der Beschranktheit des Computers werden wir an die Grenzen allen Wissens gefuhrt. Grenzen, die den Menschen beflugeln, das Moegliche weiter zu verbessern und selbst aus dem Unmoeglichen Nutzen zu ziehen. Eine brillante tour de force mit uberraschenden Aspekten, die den Leser - ob vorgebildeter Laie oder Fachkundiger - von der ersten bis zur letzten Seite fesselt.
Wahrend einer Konferenz zum "Jiidischen Nietzscheanismus" 1995 in Greifs wald hatte mich EGBERT BRIESKORN eingeladen, in der Edition der Gesam melten Werke FELIX HAUSDORFFS dessen philosophische Schriften mit einer Einleitung herauszugeben. FELIX HAUSDORFF hatte darin eng an NIETZSCHE angeschlossen, und er hatte in Greifswald sein erstes Ordinariat fUr Mathematik erhalten - ich sagte spontan und, wie sich bald herausstellen soUte, leichtsinnig ja. Statt nur mit einer kurzen Einleitung hatte ich es bald auch mit langwieri gen Erschlief&ungen des Werks und seiner Kommentierung zu tun. Doch je mehr ich mich in FELIX HAUSDORFFS Schriften einarbeitete, desto mehr notigten sie mir Respekt ab: in ihrer Klarheit, ihrer Redlichkeit, ihrer vornehmen Beschei denheit, ihrer gedanklichen Selbstandigkeit und vor allem in ihrer erstaunlichen Aktualitat. Vielleicht ist nach iiber hundert Jahren nun die Zeit gekommen, in der sie fiir die philosophische Orientierung so fruchtbar werden konnen, wie sie es verdienen. Bei der Kommentierung haben viele helfende Hande mitgewirkt. Mein Dank gilt zuerst den studentischen und wissenschaftlichen Hilfskraften: MIRKO GRON DER und KATRIN STELTER haben die Hauptarbeit in der Recherchierung der Belege iibernommen, JUDITH KARLA und TANJA SCHMIDT eine Vielzahl von Nachweisen beigesteuert, WOLFGANG SCHNEIDER und RALF WITZLER an den Vorarbeiten mitgewirkt. Doz. Dr. REINHARD PESTER (friiher Greifswald, jetzt Berlin) hat uns bei den Nachweisen zu LOTZE, Prof. Dr. MARTIN HOSE (frii her Greifswald, jetzt Miinchen) bei Zitaten aus der griechischen Literatur, Prof. Dr. GISELA FEBEL (friiher Stuttgart, jetzt Bremen) bei Zitaten aus der franzosischen Literatur, Prof. Dr. WALTER ERHART, Prof. Dr."
H. Soubies-Camy: L alg bre logique appliqu e aux techniques binaires, I parte: lezioni.- H. Soubies-Camy: L alg bre logique appliqu e aux techniques binaires, II parte: disegni.- J. Piesch: Switching Algebra.- J.P. Roth: Una teoria per la progettazione logica dei Meccanismi Automatici.
Das Unendliche hat wie keine andere Frage von jeher so tief das Gemut der Menschen bewegt," das Unendliche hat wie kaum eine andere Idee auf den Verstand so an- regend und fruchtbar gewirkt," das Unendliche ist aber auch wie kein anderer Begriff so der Aufklarung bedurftig. HILBERT [226, p. 163] Etwas mehr als 100 Jahre sind vergangen, seit in den Mathemati- schen Annalen der sechste und letzte Teil von CANTORS fundamenta- ler Arbeit UEber unendliche lineare Punktmannichfaltigkeiten erschie- nen ist. Damit war die Mengenlehre geboren und mit ihr eine prinzipiell neue Auffassung des Unendlichen in der Mathematik, verkoerpert in CANTORS Theorie der transfiniten Zahlen. Diese Theo- rie hat HILBERT als "die bewundernswerteste Blute mathematischen Geistes und uberhaupt eine der hoechsten Leistungen rein verstandes- massiger menschlicher Tatigkeit" bezeichnet. Anfangs unbeachtet oder abgelehnt, zu Ende des vorigen Jahrhunderts zunehmend anerkannt und verwendet, durch die Ent- deckung der Antinomien erneut erschuttert, ist die Mengenlehre in ihrer heutigen axiomatisierten Gestalt eines der Fundamente der Mathematik. Die Tatsache, dass alle mathematischen Begriffe auf mengentheoretische Begriffe zuruckgefuhrt werden koennen, hat ei- nige Autoren sogar zu der Behauptung veranlasst, die gesamte Ma- thematik sei letztendlich mit der Mengenlehre identisch. Wenn uns allerdings eine solche Ansicht als eine ungerechtfertigte UEberbeto- nung des Formalen gegenuber dem Inhaltlichen erscheint, so ist doch unbestritten, dass die mengentheoretische Durchdringung der Mathematik neben der Entstehung des strukturellen Denkens und der Verwendung der axiomatischen Methode ein Wesenszug der mo- dernen Mathematik ist. Das hat in zahlreichen Landern bis in den Schulunterricht hinein gewirkt.
Logic is now widely recognized to be one of the foundational disciplines of computing, and its applications reach almost every aspect of the subject, from software engineering and hardware to programming languages and artificial intelligence research. The Handbook of Logic in Computer Science is a six volume, internationally authored work which offers a comprehensive treatment of the application of the concepts of logic to theoretical computer science. Each volume is comprised of an average of five 100-page monographs and presents an in-depth overview of a major subject area. The first two volumes, available now, cover the background to the subject in terms of mathematical and computational structures. Future volumes will cover semantic structures, semantic modelling, theoretical methods in specification and verification, and logical methods in computer science. The result of five years of cooperative effort by some of the field's most eminent scholars, this series will undoubtedly be the standard reference work in logic and theoretical computer science for years to come.
Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der fonnalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet fUr die Logik eine ahnliche Fonnel- sprache an, wie sie zum Ausdruck mathematischer Beziehungen schon seit langem gebrauchlich ist. In der Mathematik wurde es heute als eine Utopie gelten, wollte man beim Aufbau einer mathematischen Disziplin sich nur der gewohnlichen Sprache bedienen. Die groBen Fortschritte, die in der Mathematik seit der Antike gemacht worden sind, sind zum wesentlichen Teil mit dadurch bedingt, daB es gelang, einen brauchbaren und leistungsfahigen Fonnalismus zu finden. - Was durch die Formel- sprache in der Mathematik erreicht wird, das solI auch in der theoretischen Logik durch diese erzielt werden, namlich eine exakte, wissenschaftliche Behandlung ihres Gegenstandes. Die logischen Sachverhalte, die zwischen Urteilen, Begriffen usw. bestehen, finden ihre Darstellung durch Formeln, deren Interpretation frei ist von den Unklarheiten, die beim sprachlichen Ausdruck leicht auftreten konnen. Der Dbergang zu logischen Folgerungen, wie er durch das SchlieBen geschieht, wird in seine letzten Elemente zerlegt und erscheint als fonnale Umgestaltung der Ausgangsfonneln nach gewissen Regeln, die den Rechenregeln in der Algebra analog sind; das logische Denken findet sein Abbild in einem LogikkalkUl. Dieser Kalkiil macht die erfolgreiche Inangriffnahme von Problemen moglich, bei denen das rein inhaltliche Denken prinzipiell versagt. Zu diesen gehort z. B.
Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Godel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to PI11-CA0. Ordinal analysis and the (Schwichtenberg-Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and PI11-CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.
Long ago, when Alexander the Great asked the mathematician Menaechmus for a crash course in geometry, he got the famous reply There is no royal road to mathematics. Where there was no shortcut for Alexander, there is no shortcut for us. Still, the fact that we have access to computers and mature programming languages means that there are avenues for us that were denied to the kings and emperors of yore. The purpose of this book is to teach logic and mathematical reasoning in practice, and to connect logical reasoning with computer programming in Haskell. Haskell emerged in the 1990s as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvelous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures. This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book, the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others. This is the updated, expanded, and corrected second edition of a much-acclaimed textbook. Praise for the first edition: Doets and van Eijck s The Haskell Road to Logic, Maths and Programming is an astonishingly extensive and accessible textbook on logic, maths, and Haskell. Ralf Laemmel, Professor of Computer Science, University of Koblenz-Landau
This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians.
This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume II, on formal (ZFC) set theory, incorporates a self-contained 'chapter 0' on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques will provide the reader with a solid foundation in set theory and provides a context for the presentation of advanced topics such as absoluteness, relative consistency results, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing.
Bringing together over twenty years of research, this book gives a complete overview of independence-friendly logic. It emphasizes the game-theoretical approach to logic, according to which logical concepts such as truth and falsity are best understood via the notion of semantic games. The book pushes the paradigm of game-theoretical semantics further than the current literature by showing how mixed strategies and equilibria can be used to analyze independence-friendly formulas on finite models. The book is suitable for graduate students and advanced undergraduates who have taken a course on first-order logic. It contains a primer of the necessary background in game theory, numerous examples and full proofs.
Der Autor vermittelt logisches Grundwissen, fundamentale Beweisprinzipien und Methoden der Mathematik. Dabei geht er u. a. folgenden Fragen nach: Was unterscheidet endliche von unendlichen Mengen? Wie lassen sich die ganzen, rationalen und reellen Zahlen aus den nat rlichen Zahlen konstruieren? Welche grundlegenden topologischen Eigenschaften besitzt die Menge der reellen Zahlen? Lassen sich die nat rlichen oder reellen Zahlen vollst ndig axiomatisch beschreiben? Pflichtlekt re f r alle Studierenden der Mathematik, Physik und Informatik.
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.
These pun-studded fables by a popular science writer make complicated mathematical concepts accessible and fun. Twelve essays take a playful approach to mathematics, investigating the topology of a warm blanket, the odds of beating a superior tennis player, and how to distinguish between fact and fallacy. 1991 edition.
Petri-Netze sind das meist beachtete und am besten untersuchte Modell fur nebenlaufige, parallele Rechnungen. In diesem Lehrbuch werden zum ersten Mal zahlreich Resultate der Originalliteratur uber Unmoglichkeiten, Moglichkeiten und die Komplexitat der Ausdrucksmittel von Petri-Netzen didaktisch aufgearbeitet und im Detail einer breiteren Leserschaft vorgestellt. Alle fur die Beweise notwendigen Techniken und mathematischen Begriffe werden erlautert. Damit wendet sich das Buch sowohl an Studierende als auch an Lehrende und Forscher. Der Inhalt konzentriert sich neben einer Darstellung der Grundbegriffe und deren Zusammenhange insbesondere auf einen Algorithmus fur die Erreichbarkeitsfrage, die Ausdrucksfahigkeit verschiedener Berechnungsbegriffe, ausgewahlte Fragen zur Entscheidbarkeit und Komplexitat, sowie Petri-Netz Semantiken mittels Sprachen und partiell geordneten Mengen und deren algebraische Charakterisierung."
Gli insegnanti si trovano in difficoltA a proposito dello spazio e della (TM)enfasi da dare agli argomenti di teoria degli insiemi, nella propria preparazione e nel proprio lavoro, perchA(c) all'universitA non A] stata loro fornita una conoscenza adeguata. Si puA tranquillamente affermare, sulla base di molta esperienza, che il matematico medio, anche chi fa ricerca, non sa cosa sia la teoria degli insiemi. Due pregiudizi si frappongono a una buona conoscenza della teoria: uno, di tipo minimalista, A] la sua identificazione con una non meglio precisata "insiemistica," un linguaggio austero fin troppo impegnativo ove lo si voglia imporre prematuramente; la (TM)altro A] di tipo massimalista e consiste nel supposto, ed effettivo legame con le questioni piA sottili dei fondamenti della matematica. Ma la teoria ha un contenuto matematico importante, e con molti risvolti di interesse didattico. Si puA dire in una parola che A] lo studio della (TM)infinito, il che comporta anche per complemento che sia uno studio del finito. Attraverso gli insiemi numerabili ed effettivamente generati si stabilisce anche un collegamento con la piA concreta teoria della calcolabilitA . Il libro A] solo una guida, non un manuale: sono indicati gli argomenti di maggior rilievo; sono offerti commenti sui risultati piA significativi; sono segnalati anche temi da non approfondire, pur conoscendone la (TM)esistenza; sono presentate con dettagli formali poche dimostrazioni, tipiche dello stile della materia; sono proposti, come istruzioni per la (TM)uso, alcuni esercizi che potrebbero essere presentarti anche a studenti delle scuole secondarie.
Alfred Tarski, one of the greatest logicians of all time, is widely thought of as 'the man who defined truth'. His mathematical work on the concepts of truth and logical consequence are cornerstones of modern logic, influencing developments in philosophy, linguistics and computer science. Tarski was a charismatic teacher and zealous promoter of his view of logic as the foundation of all rational thought, a bon-vivant and a womanizer, who played the 'great man' to the hilt. Born in Warsaw in 1901 to Jewish parents, he changed his name and converted to Catholicism, but was never able to obtain a professorship in his home country. A fortuitous trip to the United States at the outbreak of war saved his life and turned his career around, even while it separated him from his family for years. By the war's end he was established as a professor of mathematics at the University of California, Berkeley. There Tarski built an empire in logic and methodology that attracted students and distinguished researchers from all over the world. From the cafes of Warsaw and Vienna to the mountains and deserts of California, this first full length biography places Tarski in the social, intellectual and historical context of his times and presents a frank, vivid picture of a personally and professionally passionate man, interlaced with an account of his major scientific achievements.
This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with K??nig's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis.
Mathematics and logic have been central topics of concern since the
dawn of philosophy. Since logic is the study of correct reasoning,
it is a fundamental branch of epistemology and a priority in any
philosophical system. Philosophers have focused on mathematics as a
case study for general philosophical issues and for its role in
overall knowledge- gathering. Today, philosophy of mathematics and
logic remain central disciplines in contemporary philosophy, as
evidenced by the regular appearance of articles on these topics in
the best mainstream philosophical journals; in fact, the last
decade has seen an explosion of scholarly work in these areas.
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