This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.
Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.
Translated from the French, this book is an introduction to first-order model theory. Starting from scratch, it quickly reaches the essentials, namely, the back-and-forth method and compactness, which are illustrated with examples taken from algebra. It also introduces logic via the study of the models of arithmetic, and it gives complete but accessible exposition of stability theory.
The book offers a direct and up-to-date introduction to the theory of one-parameter semigroups of linear operators on Banach spaces. The book is intended for students and researchers who want to become acquainted with the concept of semigroups.
This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.
يتناول هذا المؤلف من جديد ـ بشكل أكثر دقة وتصميماً ـ مادة مُدرَّسة بجامعة بيار وماري كوري على مستوى البكالريوس، وهو يفترض معرفة العناصر الأساسية من الطوبولوجيا العامة والتكامل الحسابي والتفاضلي. يتعرض الجزء الأول من الكتاب (الفصول 1-7) إلى جوانب (مجردة) من التحليل الدالي، أما الجزء الثاني من المادة (الفصول 8-10) فيتعلق بدراسة فضاءات دالية (ملموسة) مستعملة في نظرية المعادلات التفاضلية الجزئية، تبين كيف يمكن لمبرهنات وجود(مجردة) أن تسهم في حل معادلات تفاضلية جزئية. هناك ارتباط وثيق بين هذين الفرعين من التحليل: تاريخياً، تطور التحليل الدالي(المجرد) ليجيب عن أسئلة أثيرت عند حل المعادلات التفاضلية الجزئية، وفي المقابل أدى تطور التحليل الدالي (المجرد) إلى تحفيز كبير لنظرية المعادلات التفاضلية الجزئية. سيكون هذا الكتاب مفيداً لكل من الطلبة المهتمين بالرياضيات البحثية، وكذا أولئك المهتمين بالتوجه نحو الرياضيات التطبيقية. العبيكان للنشر