(Reprint of the 1967 edition)

Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

Frege is widely regarded as having set much of the agenda of contemporary analytic philosophy. As standardly read, he meant to introduce-and make crucial contributions to-the project of giving an account of the workings of (an improved version of) natural language. Yet, despite the great admiration most contemporary philosophers feel for Frege, it is widely believed that he committed a large number of serious, and inexplicable, blunders. For, if Frege really meant to be constructing a theory of the workings of (some version of) natural language, then a significant number of his stated views-including views that he claimed to be central to his philosophical picture-are straightforwardly wrong. But did Frege mean to be giving an account of the workings of language? He himself never actually claimed to be doing this, and, indeed, never even described such a project. Taking Frege at his Word offers an interpretation that is based on a different approach to his writings. Rather than using the contributions he is taken to have made to contemporary work in the philosophy of language to infer what his projects were, Joan Weiner gives priority to Frege's own accounts of what he means to be doing. She provides a very different view of Frege's project. One might suspect that, on such a reading, Frege's writings would have purely antiquarian interest, but this would be a mistake. The final two chapters show that Frege offers us new ways of addressing some of the philosophical problems that worry us today.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In "Higher Topos Theory," Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.

The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

An introduction to applying predicate logic to testing and verification of software and digital circuits that focuses on applications rather than theory. Computer scientists use logic for testing and verification of software and digital circuits, but many computer science students study logic only in the context of traditional mathematics, encountering the subject in a few lectures and a handful of problem sets in a discrete math course. This book offers a more substantive and rigorous approach to logic that focuses on applications in computer science. Topics covered include predicate logic, equation-based software, automated testing and theorem proving, and large-scale computation. Formalism is emphasized, and the book employs three formal notations: traditional algebraic formulas of propositional and predicate logic; digital circuit diagrams; and the widely used partially automated theorem prover, ACL2, which provides an accessible introduction to mechanized formalism. For readers who want to see formalization in action, the text presents examples using Proof Pad, a lightweight ACL2 environment. Readers will not become ALC2 experts, but will learn how mechanized logic can benefit software and hardware engineers. In addition, 180 exercises, some of them extremely challenging, offer opportunities for problem solving. There are no prerequisites beyond high school algebra. Programming experience is not required to understand the book's equation-based approach. The book can be used in undergraduate courses in logic for computer science and introduction to computer science and in math courses for computer science students.

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging uses of model theory have created a highly fragmented literature. On the one hand, many philosophically significant results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a serious background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are scattered across disconnected pockets of papers. The first aim of this book, then, is to explore the philosophical uses of model theory, focusing on the central topics of reference, realism, and doxology. Its second aim is to address important questions in the philosophy of model theory, such as: sameness of theories and structure, the boundaries of logic, and the classification of mathematical structures. Philosophy and Model Theory will be accessible to anyone who has completed an introductory logic course. It does not assume that readers have encountered model theory before, but starts right at the beginning, discussing philosophical issues that arise even with conceptually basic model theory. Moreover, the book is largely self-contained: model-theoretic notions are defined as and when they are needed for the philosophical discussion, and many of the most philosophically significant results are given accessible proofs.

Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, "The Calculi of Lambda-Conversion" (1941), established an invaluable tool that computer scientists still use today.

Even beyond the accomplishment of that book, however, his second Princeton book, "Introduction to Mathematical Logic," defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic.

Church was one of the principal founders of the Association for Symbolic Logic; he founded the "Journal of Symbolic Logic" in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of recent developments, and the rest of the papers are chosen to complement the invited talks. This volume includes surveys, tutorials, and selected research papers from the 2004 meeting. Highlights include a tutorial survey of the recent highpoints of universal algebra, written by a leading expert; explorations of foundational questions; and a quartet of model theory papers giving an excellent reflection of current work in model theory, from the most abstract aspect 'abstract elementary classes' to issues around p-adic integration.

In 1931 Kurt Godel published his fundamental paper, "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Godel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences--perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Godel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

Marking the 50th anniversary of the original publication of Godel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling and most frequently translated books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

Gli autori, basandosi sulla loro esperienza di ricerca, propongono in due volumi un testo di riferimento per acquisire una solida formazione specialistica nella logica.Nei due volumi vengono presentati in maniera innovativa e rigorosa temi di logica tradizionalmente affrontati nei corsi universitari di secondo livello. Questo primo volume e dedicato ai teoremi fondamentali sulla logica del primo ordine e alle loro principali conseguenze. Il testo e rivolto in particolare agli studenti dei corsi di laurea magistrale.

Neil Tennant presents an original logical system with unusual philosophical, proof-theoretic, metalogical, computational, and revision-theoretic virtues. Core Logic, which lies deep inside Classical Logic, best formalizes rigorous mathematical reasoning. It captures constructive relevant reasoning. And the classical extension of Core Logic handles non-constructive reasoning. These core systems fix all the mistakes that make standard systems harbor counterintuitive irrelevancies. Conclusions reached by means of core proof are relevant to the premises used. These are the first systems that ensure both relevance and adequacy for the formalization of all mathematical and scientific reasoning. They are also the first systems to ensure that one can make deductive progress with potential logical strengthening by chaining proofs together: one will prove, if not the conclusion sought, then (even better!) the inconsistency of one's accumulated premises. So Core Logic provides transitivity of deduction with potential epistemic gain. Because of its clarity about the true internal structure of proofs, Core Logic affords advantages also for the automation of deduction and our appreciation of the paradoxes.

Not everything is black and white. Our daily lives are full of vagueness or fuzziness. Language is the most obvious example - for instance, when we describe someone as tall, it is as though there is a particular height beyond which a person can be considered 'tall'. Likewise the terms 'blond' or 'overweight' in common usage. We often think in discontinuous categories when we are considering something continuous. In this book, van Deemter cuts across various disciplines in considering the nature and importance of vagueness. He looks at the principles of measurement, and how we choose categories; the vagueness lurking behind what seems at first sight crisp concepts such as that of the biological 'species'; uncertainties in grammar and the impact of vagueness on the programmes of Chomsky and Montague; vagueness and mathematical logic; computers, vague descriptions, and Natural Language Generation in AI (a new class of programs will allow computers to handle descriptions such as 'the man in the yellow shirt'). Van Deemter shows why vagueness is in various circumstances both unavoidable and useful, and how we are increasingly able to handle fuzziness in mathematical logic and computer science.

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability. Both the basics and recent research results are desribed, providing a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Goedel's completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.

H. Hermes: Basic notions and applications of the theory of decidability.- D. Kurepa: On several continuum hypotheses.- A. Mostowski: Models of set theory.- A. Robinson: Problems and methods of model theory.- S. Sochor, B. Balcar: The general theory of semisets. Syntactic models of the set theory.

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and modern-day programming. This book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts:
Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words.
Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle.
Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build.
Part IV" Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows.
Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, thetext includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web.

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential
and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and modern-day programming. This book concentrates on using logic as a tool: making and using formal proofs and
disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts:
Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words.
Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle.
Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build.
Part IV" Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows.
Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, thetext includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the
web.

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.

Wie ist ein Ring definiert, wann kann man Grenzprozesse vertauschen, was sind lineare Ordnungen und wozu benoetigt man das Zornsche Lemma in der Linearen Algebra? Das Buch will seinen Lesern helfen, sich in der Fulle der grundlegenden mathematischen Definitionen zurecht zu finden und exemplarische mathematische Ergebnisse einordnen und ihre Eigenheiten verstehen zu koennen. Es behandelt hierzu je zwoelf Schlusselkonzepte der folgenden zwoelf Themengebiete der Mathematik: Grundlagen Zahlen Zahlentheorie Diskrete Mathematik Lineare Algebra Algebra Elementare Analysis Hoehere Analysis Topologie und Geometrie Numerik Stochastik Mengenlehre und Logik Ein besonderes Augenmerk liegt auf einer knappen und prazisen, dabei aber nicht zu formalen Darstellung. Dadurch erlauben die einzelnen Beitrage ein fokussiertes Nachlesen ebenso wie ein neugieriges Kennenlernen. Das Buch ist geschrieben fur Studierende der Mathematik ab dem ersten Semester und moechte ein treuer Begleiter und eine zuverlassige Orientierungshilfe fur das gesamte Studium sein. Die 2. Auflage ist vollstandig durchgesehen und um Literaturangaben erganzt.

"Deduction" is an efficient and elegant presentation of classical first-order logic. It presents a truth tree system based on the work of Jeffrey, as well as a natural deduction system inspired by that of Kalish and Montague. Both are very natural and easy to learn. The definition of a formula excludes free variables, and the deduction system uses "Show" lines; the combination allows rules to be stated very simply.

The book's main innovation is its final part, which contains chapters on extensions and revisions of classical logic: modal logic, many-valued logic, fuzzy logic, intuitionistic logic, counterfactuals, deontic logic, common-sense reasoning, and quantified modal logic. These have been areas of great logical and philosophical interest over the past 40 years, but few other textbooks treat them in any depth. "Deduction" makes these areas accessible to introductory students. All chapters have discussions of the underlying semantics and present both truth tree and deduction systems.

New features in this edition, in addition to truth tree systems for classical and nonclassical logics, include new and simpler rules for modal logic, deontic logic, and counterfactuals; discussions of many-valued, fuzzy, and intuitionistic logics; an introduction to common-sense reasoning (nonmonotonic logic); and extensively reworked problem sets, designed to lead students gradually from easier to more difficult problems. This new edition also features web-based programs that make use of the book's methods. Each program is set up to give students symbolization problems, give them hints, grade their work, and do problems for them.