Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

    The author of What Is the Name of This Book? presents a compilation of more than two hundred challenging new logic puzzles--ranging from simple brainteasers to complex mathematical paradoxes.

    His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

    There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with

    Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O. Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources. Moreover, a number of people have assisted in our work by pro- viding helpful comments on portions of the manuscript. In this respect, we extend our hearty thanks in particular to P. Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel.

    Debate is the investigative technique used in Tibetan education to sharpen analytical capacities and convey philosophical concepts, so it is essential to master its procedure. Using a debate manual by Pur-bu-jok Jam-ba-gya-tso (1825–1901) as his basis, Daniel Perdue covers elementary debate and demonstrates its application to a variety of secular and religious educational contexts. The translation is supplied with annotations on procedure and content drawn from Tibetan teachers expert in debate.

    It deals with two important themes of Buddhism: emptiness and dependent arising.

    Despite his importance in the history of philosophy, a unified statement of his thought has been unavailable. With this publication, readers at long last are offered the philosopher's only known, complete, and coherent account of his own work. Originally delivered as the Cambridge Conferences Lectures of 1898, Reasoning and the Logic of Things is the most accessible and thorough introduction to Peirce's mature thought to be found within the compass of a single book.Beginning with an explanation of the nature of philosophy, Peirce proceeds to illustrate his claim that mathematics provides the foundation of our logic and metaphysics. We find here the clearest formulation of an idea present in Peirce's thought since the 1860s, the distinction between three kinds of reasoning: induction, deduction, and retroduction. Then follows an introduction to Peirce's chief logical doctrines, as well as his attempts to provide a classification of the sciences, a theory of categories, and a theory of science. In conclusion, turning from "reasoning" to the "logic of things," Peirce called for an evolutionary cosmology to explain the reality of laws and described the kinds of reasoning he employed in developing this cosmology.At the urging of his friend William James, Peirce made an uncharacteristic effort in these lectures to present his ideas in terms intelligible to a general audience--those without advanced training in logic and philosophy. The introductory materials by Kenneth Ketner and Hilary Putman add to the volume's lucidity. Consequently, this book will be a valuable source for readers outside of the circle of Peirce specialists.

    Taking advantage of Tarski's World 4.0, the text skilfully balances the semantic conception of logic with methods of proof. The book contains eleven chapters, in four parts. Part I is about propositional logic, Part II about quantifier logic. Part III contains chapters on set theory and inductive definitions. Part IV contains advanced topics in logic, including topics of importance in applications of logic in computer science. The Language of First-order Logic contains hundreds of problems and exercises for the user to work through.

    Reading level: grade 5; ability level: grade 7-adult; 32 activities and answers included. Reproducible for single-classroom or single-home use.