Hilbert's Tenth Problem
An Introduction to Logic, Number Theory, and Computability
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-4399-3 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-4399-3 (ISBN)
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900. It asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. This book is an exposition of their achievement.
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists.
This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Godel's incompleteness theorems.
Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists.
This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Godel's incompleteness theorems.
Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
M. Ram Murty, Queen's University, Kingston, ON, Canada. Brandon Fodden, Carleton University, Ottawa, ON, Canada.
Introduction
Cantor and infinity
Axiomatic set theory
Elementary number theory
Computability and provability
Hilbert's tenth problem
Applications of Hilbert's tenth problem
Hilbert's tenth problem over number fields
Background material
Bibliography
Index
| Erscheinungsdatum | 02.07.2019 |
|---|---|
| Reihe/Serie | Student Mathematical Library |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 140 x 216 mm |
| Gewicht | 313 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre | |
| ISBN-10 | 1-4704-4399-6 / 1470443996 |
| ISBN-13 | 978-1-4704-4399-3 / 9781470443993 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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Buch | Softcover (2024)
Springer (Verlag)
CHF 41,95
