Author: Richard Kaye
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 312
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Book Description
Non-standard models of arithmetic are of interest to mathematicians through the presence of infinite integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s, they have come to play an important role in model theory, and in combinatorics through independence results such as the Paris-Harrington theorem. This book is an introduction to these developments, and stresses the interplay between the first-order theory, recursion-theoretic aspects, and the structural properties of these models. Prerequisites for an understanding of the text have been kept to a minimum, these being a basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets. Consequently, the book is suitable for postgraduate students coming to the subject for the first time, and a number of exercises of varying degrees of difficulty will help to further the reader's understanding.
Author: Ali Enayat
Publisher: American Mathematical Soc.
ISBN: 0821835351
Category : Mathematics
Languages : en
Pages : 184
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Book Description
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.
Author: Matthew Katz
Publisher: American Mathematical Soc.
ISBN: 1470442906
Category : Mathematics
Languages : en
Pages : 224
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Book Description
This book takes the reader on a journey through Ramsey theory, from graph theory and combinatorics to set theory to logic and metamathematics. Written in an informal style with few requisites, it develops two basic principles of Ramsey theory: many combinatorial properties persist under partitions, but to witness this persistence, one has to start with very large objects. The interplay between those two principles not only produces beautiful theorems but also touches the very foundations of mathematics. In the course of this book, the reader will learn about both aspects. Among the topics explored are Ramsey's theorem for graphs and hypergraphs, van der Waerden's theorem on arithmetic progressions, infinite ordinals and cardinals, fast growing functions, logic and provability, Gödel incompleteness, and the Paris-Harrington theorem. Quoting from the book, “There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time. Ramsey theory is one of those ladders.”
Author: Lou van den Dries
Publisher: Springer
ISBN: 3642549365
Category : Mathematics
Languages : en
Pages : 201
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Book Description
Presenting recent developments and applications, the book focuses on four main topics in current model theory: 1) the model theory of valued fields; 2) undecidability in arithmetic; 3) NIP theories; and 4) the model theory of real and complex exponentiation. Young researchers in model theory will particularly benefit from the book, as will more senior researchers in other branches of mathematics.
Author: Edward Nelson
Publisher: Princeton University Press
ISBN: 1400858925
Category : Mathematics
Languages : en
Pages : 199
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Book Description
This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Author: Lyn D. English
Publisher: Routledge
ISBN: 0805814574
Category : Mathematics
Languages : en
Pages : 365
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Book Description
First Published in 1995. Routledge is an imprint of Taylor & Francis, an informa company.
Author: Harold M. Edwards
Publisher: American Mathematical Soc.
ISBN: 9780821844397
Category : Mathematics
Languages : en
Pages : 228
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Book Description
Among the topics featured in this textbook are: congruences; the fundamental theorem of arithmetic; exponentiation and orders; primality testing; the RSA cipher system; polynomials; modules of hypernumbers; signatures of equivalence classes; and the theory of binary quadratic forms. The book contains exercises with answers.
Author: Houghton Mifflin Harcourt
Publisher: Houghton Mifflin School
ISBN: 9789812806604
Category : Education
Languages : en
Pages : 136
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Book Description
This monograph serves as a resource book on the Model Method. The main purpose is to make explicit how the Model Method is used to develop students' understanding of fundamental mathematics concepts and proficiency in solving basic mathematics word problems. Through the construction of a pictorial model to represent the known and unknown quantities and their relationships in a problem, students gain better understanding of the problem and develop their abilities in mathematical thinking and problem solving. This would provide a strong foundation for the learning of matheatics from the primary to secondary levels and beyond. This monograph also features the Mathematics Framework of the Singapore mathematics curriculum, and discusses the changes that it has undergone over the past two decades. These changes reflected the changing emphases, needs and challenges in the mathematics curriculum as we entered the 21s century.
Author: Christopher James Phillips
Publisher: University of Chicago Press
ISBN: 022618496X
Category : Education
Languages : en
Pages : 242
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Book Description
An era of sweeping cultural change in America, the postwar years saw the rise of beatniks and hippies, the birth of feminism, and the release of the first video game. This book examines the rise and fall of the new math as a marker of the period's political and social ferment.
Author: Orrin H. Pilkey
Publisher: Columbia University Press
ISBN: 0231506996
Category : Science
Languages : en
Pages : 397
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Book Description
Noted coastal geologist Orrin Pilkey and environmental scientist Linda Pilkey-Jarvis show that the quantitative mathematical models policy makers and government administrators use to form environmental policies are seriously flawed. Based on unrealistic and sometimes false assumptions, these models often yield answers that support unwise policies. Writing for the general, nonmathematician reader and using examples from throughout the environmental sciences, Pilkey and Pilkey-Jarvis show how unquestioned faith in mathematical models can blind us to the hard data and sound judgment of experienced scientific fieldwork. They begin with a riveting account of the extinction of the North Atlantic cod on the Grand Banks of Canada. Next they engage in a general discussion of the limitations of many models across a broad array of crucial environmental subjects. The book offers fascinating case studies depicting how the seductiveness of quantitative models has led to unmanageable nuclear waste disposal practices, poisoned mining sites, unjustifiable faith in predicted sea level rise rates, bad predictions of future shoreline erosion rates, overoptimistic cost estimates of artificial beaches, and a host of other thorny problems. The authors demonstrate how many modelers have been reckless, employing fudge factors to assure "correct" answers and caring little if their models actually worked. A timely and urgent book written in an engaging style, Useless Arithmetic evaluates the assumptions behind models, the nature of the field data, and the dialogue between modelers and their "customers."
