This book provides an introduction to Gödel's theorem. Gödel's theorem states that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic. The content of the theorem is elucidated and distinguished from that of other claims with which it is often confused. The significance of the theorem is also discussed. Particular emphasis is laid on the appeal of axiomatization and on attempts that were made, in the half century preceding Gödel's proof, to provide the very thing that the theorem precludes. This includes discussion of Hilbert's programme, part of which was to provide a consistent foundation for mathematics and to demonstrate its consistency by mathematical means. Two proofs of Gödel's theorem are given. The second and more elaborate proof is also shown to yield Gödel's second theorem: that no consistent axiomatization of arithmetic can be used to prove a statement corresponding to a statement of its own consistency. The final two chapters of the book explore the implications of Gödel's results: for Hilbert's programme; for the question whether the human mind, in its capacity to think beyond any given axiomatization of arithmetic, has powers beyond those of any possible computer; and for the nature of mathematics.

Very Short Introductions: Brilliant, Sharp, Inspiring Kurt Gödel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Gödel's Theorem has since established itself as a landmark intellectual achievement, having a profound impact on today's mathematical ideas. Gödel and his theorem have attracted something of a cult following, though his theorem is often misunderstood. This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states. A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content. Moore also discusses the most important philosophical implications of the theorem. In particular, Moore addresses the famous question of whether the theorem shows the human mind to have mathematical powers beyond those of any possible computer ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Peter Smith examines Gödel's Theorems, how they were established and why they matter.

Logic is often perceived as having little to do with the rest of philosophy, and even less to do with real life. Graham Priest explores the philosophical roots of the subject, explaining how modern formal logic addresses many issues.

"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.

Mathematics is a fundamental human activity that can be practised and understood in a multitude of ways; indeed, mathematical ideas themselves are far from being fixed, but are adapted and changed by their passage across periods and cultures. In this Very Short Introduction, Jacqueline Stedall explores the rich historical and cultural diversity of mathematical endeavour from the distant past to the present day. Arranged thematically, to exemplify the varied contexts in which people have learned, used, and handed on mathematics, she also includes illustrative case studies drawn from a range of times and places, including early imperial China, the medieval Islamic world, and nineteenth-century Britain. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Graham Priest shows that formal logic is a powerful, exciting part of modern philosophy -- a tool for thinking about everything from the existence of God and the reality of time to paradoxes of probability. Explaining formal logic in simple, non-technical terms, this edition includes new sections on mathematical algorithms, axioms, and proofs.