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Author | : William S. Hatcher |
Publisher | : Elsevier |
Total Pages | : 331 |
Release | : 2014-05-09 |
Genre | : Mathematics |
ISBN | : 1483189635 |
Download The Logical Foundations of Mathematics Book in PDF, ePub and Kindle
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.
Author | : Kenneth Kunen |
Publisher | : |
Total Pages | : 251 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 9781904987147 |
Download The Foundations of Mathematics Book in PDF, ePub and Kindle
Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.
Author | : Luca Incurvati |
Publisher | : Cambridge University Press |
Total Pages | : 255 |
Release | : 2020-01-23 |
Genre | : History |
ISBN | : 1108497829 |
Download Conceptions of Set and the Foundations of Mathematics Book in PDF, ePub and Kindle
Presents a detailed and critical examination of the available conceptions of set and proposes a novel version.
Author | : Garret Sobczyk |
Publisher | : Springer Science & Business Media |
Total Pages | : 373 |
Release | : 2012-10-26 |
Genre | : Mathematics |
ISBN | : 0817683852 |
Download New Foundations in Mathematics Book in PDF, ePub and Kindle
The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner. New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.
Author | : Raymond L. Wilder |
Publisher | : Courier Corporation |
Total Pages | : 352 |
Release | : 2013-09-26 |
Genre | : Mathematics |
ISBN | : 0486276201 |
Download Introduction to the Foundations of Mathematics Book in PDF, ePub and Kindle
Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition.
Author | : Paul Taylor |
Publisher | : Cambridge University Press |
Total Pages | : 590 |
Release | : 1999-05-13 |
Genre | : Mathematics |
ISBN | : 9780521631075 |
Download Practical Foundations of Mathematics Book in PDF, ePub and Kindle
This book is about the basis of mathematical reasoning both in pure mathematics itself (particularly algebra and topology) and in computer science (how and what it means to prove correctness of programs). It contains original material and original developments of standard material, so it is also for professional researchers, but as it deliberately transcends disciplinary boundaries and challenges many established attitudes to the foundations of mathematics, the reader is expected to be open minded about these things.
Author | : Matthias Baaz |
Publisher | : Cambridge University Press |
Total Pages | : 541 |
Release | : 2011-06-06 |
Genre | : Mathematics |
ISBN | : 1139498436 |
Download Kurt Gödel and the Foundations of Mathematics Book in PDF, ePub and Kindle
This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.
Author | : M.J. Beeson |
Publisher | : Springer Science & Business Media |
Total Pages | : 484 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642689523 |
Download Foundations of Constructive Mathematics Book in PDF, ePub and Kindle
This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.
Author | : John P. Mayberry |
Publisher | : Cambridge University Press |
Total Pages | : 454 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 9780521770347 |
Download The Foundations of Mathematics in the Theory of Sets Book in PDF, ePub and Kindle
This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.
Author | : Michael D. Greenberg |
Publisher | : Courier Corporation |
Total Pages | : 660 |
Release | : 2013-01-01 |
Genre | : Mathematics |
ISBN | : 0486492796 |
Download Foundations of Applied Mathematics Book in PDF, ePub and Kindle
"A longtime classic text in applied mathematics, this volume also serves as a reference for undergraduate and graduate students of engineering. Topics include real variable theory, complex variables, linear analysis, partial and ordinary differential equations, and other subjects. Answers to selected exercises are provided, along with Fourier and Laplace transformation tables and useful formulas. 1978 edition"--