Overview

People have always been interested in numbers, in particular the natural numbers and they have an intuitive notion of what these numbers are. In the late 19th-century, mathematicians such as Grassman, Frege and Dedekind, gave definitions for these familiar objects. Since then, the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. The aim of this book by Hajek and Pudlak is to cover some of the most important results in the study of a first order theory of the natural numbers, called ""Peano arithmetic"" and its fragments (subtheories). The field is quite active, but only a small part of the results has been covered in monographs. This book is divided into three parts. In Part A, the authors develop parts of mathematics and logic in various fragments. Part B is devoted to incompleteness. Part C studies systems that have the induction schema restricted to bounded formulas (""bounded arithmetic""). One highlight of this section is the relation of provability to computational complexity. The study of formal systems for arithmetic is a prerequisite for understanding results such as Godel's theorems. This book is intended for those who want to learn more about such systems and who want to follow current research in the field. The book contains a bibliography of approximately 1000 items.

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9783540506324

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