Introduction to Model Theory

Author: Philipp Rothmaler

Publisher: CRC Press

ISBN: 9789056993139

Category: Mathematics

Page: 324

View: 2281

DOWNLOAD NOW »

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.
Release

The Logic of Infinity

Author: Barnaby Sheppard

Publisher: Cambridge University Press

ISBN: 1139952773

Category: Mathematics

Page: N.A

View: 9015

DOWNLOAD NOW »

Few mathematical results capture the imagination like Georg Cantor's groundbreaking work on infinity in the late nineteenth century. This opened the door to an intricate axiomatic theory of sets which was born in the decades that followed. Written for the motivated novice, this book provides an overview of key ideas in set theory, bridging the gap between technical accounts of mathematical foundations and popular accounts of logic. Readers will learn of the formal construction of the classical number systems, from the natural numbers to the real numbers and beyond, and see how set theory has evolved to analyse such deep questions as the status of the continuum hypothesis and the axiom of choice. Remarks and digressions introduce the reader to some of the philosophical aspects of the subject and to adjacent mathematical topics. The rich, annotated bibliography encourages the dedicated reader to delve into what is now a vast literature.
Release

Cantorian Set Theory and Limitation of Size

Author: Michael Hallett

Publisher: Oxford University Press

ISBN: 9780198532835

Category: Mathematics

Page: 343

View: 5020

DOWNLOAD NOW »

Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.
Release

Rough Sets and Current Trends in Computing

Second International Conference, RSCTC 2000 Banff, Canada, October 16-19, 2000 Revised Papers

Author: Wojciech Ziarko,Yiyu Yao

Publisher: Springer Science & Business Media

ISBN: 3540430741

Category: Computers

Page: 676

View: 6960

DOWNLOAD NOW »

This book constitutes the thoroughly refereed post-proceedings of the Second International Conference on Rough Sets and Current Trends in Computing, RSCTC 2000, held in Banff, Canada in October 2000. The 80 revised papers presented together with an introduction and three keynote presentations have gone through two rounds of reviewing and revision. The papers are organized in topical sections on granual computing, rough sets and systems, fuzzy sets and systems, rough sets and data mining, nonclassical logics and reasoning, pattern recognition and image processing, neural networks and genetic algorithms, and current trends in computing.
Release

Handbook of Philosophical Logic

Author: Dov M. Gabbay,Franz Guenthner

Publisher: Springer Science & Business Media

ISBN: 9781402019661

Category: Philosophy

Page: 323

View: 3083

DOWNLOAD NOW »

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good! The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other.
Release

Verification, Model Checking, and Abstract Interpretation

12th International Conference, VMCAI 2011, Austin, TX, USA, January 23-25, 2011 Proceedings

Author: Ranjit Jhala,David Schmidt

Publisher: Springer Science & Business Media

ISBN: 3642182747

Category: Computers

Page: 419

View: 7216

DOWNLOAD NOW »

This book constitutes the refereed proceedings of the 12th International Conference on Verification, Model Checking, and Abstract Interpretation, VMCAI 2011, held in Austin, TX, USA, in January 2011, co-located with the Symposium on Principles of Programming Languages, POPL 2011. The 24 revised full papers presented together with 4 invited talks were carefully reviewed and selected from 71 initial submissions. The papers showcases state-of-the-art research in areas such as verification, model checking, abstract interpretation and address any programming paradigm, including concurrent, constraint, functional, imperative, logic and object-oriented programming. Further topics covered are static analysis, deductive methods, program certification, debugging techniques, abstract domains, type systems, and optimization.
Release

Limits Of Mathematical Modeling In The Social Sciences, The: The Significance Of Godel's Incompleteness Phenomenon

Author: Doria Francisco Antonio

Publisher: World Scientific

ISBN: 1786343177

Category: Mathematics

Page: 288

View: 1467

DOWNLOAD NOW »

Current mathematical models are notoriously unreliable in describing the time evolution of unexpected social phenomena, from financial crashes to revolution. Can such events be forecast? Can we compute probabilities about them? Can we model them? This book investigates and attempts to answer these questions through Gödel's two incompleteness theorems, and in doing so demonstrates how influential Gödel is in modern logical and mathematical thinking. Many mathematical models are applied to economics and social theory, while Gödel's theorems are able to predict their limitations for more accurate analysis and understanding of national and international events. This unique discussion is written for graduate level mathematicians applying their research to the social sciences, including economics, social studies and philosophy, and also for formal logicians and philosophers of science.
Release

Theory and Experiment

Recent Insights and New Perspectives on Their Relation

Author: Diderik Batens,Jean-Paul van Bendegem

Publisher: Springer Science & Business Media

ISBN: 9400928750

Category: Science

Page: 296

View: 9873

DOWNLOAD NOW »

This is not "another collection of contributions on a traditional subject." Even more than we dared to expect during the preparatory stages, the papers in this volume prove that our thinking about science has taken a new turn and has reached a new stage. The progressive destruction of the received view has been a fascinating and healthy experience. At present, the period of destruction is over. A richer and more equilibrated analysis of a number of problems is possible and is being cru'ried out. In this sense, this book comes right on time. We owe a lot to the scholars of the Kuhnian period. They not only did away with obstacles, but in several respects instigated a shift in attention that changed history and philosophy of science in a irreversible way. A c1earcut example - we borrow it from the paper by Risto Hilpinen - concerns the study of science as a process, Rnd not only as a result. Moreover, they apparently reached several lasting results, e.g., concerning the tremendous impact of theoretical conceptions on empirical data. Apart from baffling people for several decades, this insight rules out an other return to simple-minded empiricism in the future.
Release