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This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.
This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
Have you ever faced a mathematical problem and had no idea how to approach it? Or perhaps you had an idea but got stuck halfway through? This book guides you in developing your creativity, as it takes you on a voyage of discovery into mathematics. Readers will not only learn strategies for solving problems and logical reasoning, but they will also learn about the importance of proofs and various proof techniques. Other topics covered include recursion, mathematical induction, graphs, counting, elementary number theory, and the pigeonhole, extremal and invariance principles. Designed to help students make the transition from secondary school to university level, this book provides readers with a refreshing look at mathematics and deep insights into universal principles that are valuable far beyond the scope of this book. Aimed especially at undergraduate and secondary school students as well as teachers, this book will appeal to anyone interested in mathematics. Only basic secondary school mathematics is required, including an understanding of numbers and elementary geometry, but no calculus. Including numerous exercises, with hints provided, this textbook is suitable for self-study and use alongside lecture courses.
This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory. The following topics are covered: • Forcing and constructability • The Solovay-Shelah Theorem i.e. the equiconsistency of ‘every set of reals is Lebesgue measurable’ with one inaccessible cardinal • Fine structure theory and a modern approach to sharps • Jensen’s Covering Lemma • The equivalence of analytic determinacy with sharps • The theory of extenders and iteration trees • A proof of projective determinacy from Woodin cardinals. Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers.
Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures
Ever since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics.
This introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.
This is the proceedings of the AMS special session on nonstandard models of arithmetic and set theory held at the Joint Mathematics Meetings in Baltimore (MD). The volume opens with an essay from Haim Gaifman that probes the concept of non-standardness in mathematics and provides a fascinating mix of historical and philosophical insights into the nature of nonstandard mathematical structures. In particular, Gaifman compares and contrasts the discovery of nonstandard models with other key mathematical innovations, such as the introduction of various number systems, the modern concept of function, and non-Euclidean geometries. Other articles in the book present results related to nonstandard models in arithmetic and set theory, including a survey of known results on the Turing upper bounds of arithmetic sets and functions. The volume is suitable for graduate students and research mathematicians interested in logic, especially model theory.