*A Critical Introduction*

Author: Michael Potter

Publisher: Clarendon Press

ISBN: 0191556432

Category: Philosophy

Page: 360

View: 7063

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## Set Theory and its Philosophy

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
## Set Theory and its Philosophy

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
## Set Theory and Its Philosophy

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
## Handbook of Set Theory

Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.
## Introduction to Axiomatic Set Theory

In 1963, the first author introduced a course in set theory at the Uni versity of Illinois whose main objectives were to cover G6del's work on the consistency of the axiom of choice (AC) and the generalized con tinuum hypothesis (GCH), and Cohen's work on the independence of AC and the GCH. Notes taken in 1963 by the second author were the taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are highlighted, and second, the student who wishes to master the sub ject is compelled to develop the details on his own. However, an in structor using a "fast development" text must devote much class time to assisting his students in their efforts to bridge gaps in the text. We have chosen instead a development that is quite detailed and complete. For our slow development we claim the following advantages. The text is one from which a student can learn with little supervision and instruction. This enables the instructor to use class time for the presentation of alternative developments and supplementary material.
## Sets

This book is an introduction to set theory in which the author develops the subject from first principles and presupposes little more than an elementary grounding in logic. Throughout much attention is paid to the historical and philosophical background which illuminates the subject's development. This book differs from most by providing a particularly elegant and intuitive approach based on Scott's formulation of standard set theory in which sets are built up stage by stage. This approach has the advantage of introducing the axioms of set theory in a natural way and shows how they come to take the form they do. The book covers all the basic tools of set theory: the natural numbers, cardinals, ordinals, and the axiom of choice in some detail. It also provides an account of the representation theory of lattices and how this is closely connected with the various forms of the axiom of choice.
## Georg Cantor

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula. Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.
## Introduction to Logic

Part I of this coherent, well-organized text deals with formal principles of inference and definition. Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Ideal for undergraduates.
## Gilles Deleuze's Philosophy of Time

This book provides an overall interpretation of Deleuze's philosophy alongside a critical introduction to one of the most important unifying ideas in his work: the construction of new and important philosophies of time.
## Philosophy and Educational Policy

What are the concepts and theories behind current debates about education? This comprehensive introduction to philosophy of education discusses issues that are of current public interest and debate. It locates education at the heart of questions concerned with culture, ethics, politics, economics and shows how key educational issues have to be approached in a contextual way. Written in a clear and accessible manner with current issues in mind the book covers: the curriculum teaching and learning educational research assessment moral, personal and civic education autonomy and multicultural issues in a liberal society education and work privatisation and markets This book will be particularly useful to students on Education Studies courses, to those preparing for a career in teaching, to students of politics and to serving teachers undertaking further study in education.
## Theories of Democracy

This is the first book to be published in this exciting new series on political philosophy. Cunningham provides a critical and clear introduction to the main contemporary approaches to democracy: participatory democracy, classic and radical pluralism, deliberative democracy, catallaxy, and others. Also discussed are theorists in the background of current democratic thought, such as Tocqueville, Mill, and Rousseau. The book includes applications of democratic theories including an extended discussion of democracy and globalisation.
## Rights

We take rights to be fundamental to everyday life. Rights are also controversial and hotly debated both in theory and practice. Where do rights come from? Are they invented or discovered? What sort of rights are there and who is entitled to them? In this comprehensive introduction, Tom Campbell introduces and critically examines the key philosophical debates about rights. The first part of the book covers historical and contemporary theories of rights, including the origin and variety of rights and standard justifications of them. He considers challenges to rights from philosophers such as Bentham, Burke and Marx. He also examines different theories of rights, such as natural law, social contract, utilitarian and communitarian theories of rights and the philosophers and political theorists associated with them, such as John Stuart Mill, John Rawls, Robert Nozick and Michael Sandel. The second part of the book explores the role of rights-promoting institutions and critically assesses legal rights and international human rights, including the United Nations. The final part of the book examines how philosophies of rights can be applied to freedom of speech, issues of social welfare and the question of self-determination for certain groups or peoples. Rights: A Critical Introduction is essential reading for anyone new to the subject of rights and any student of political philosophy, politics and law.
## Computability and Logic

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual.
## Philosophy of Mathematics

Philosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author?s personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals
## Multiculturalism

What is multiculturalism and what are the different theories used to justify it? Are multicultural policies a threat to liberty and equality? Can liberal democracies accommodate minority groups without sacrificing peace and stability? In this clear introduction to the subject, Michael Murphy explores these questions and critically assesses multiculturalism from the standpoint of political philosophy and political practice. The book explores the origins and contemporary usage of the concept of multiculturalism in the context of debates about citizenship, egalitarian justice and conflicts between individual and collective rights. The ideas of some of the most influential champions and critics of multiculturalism, including Will Kymlicka, Chandran Kukathas, Susan Okin and Brian Barry, are also clearly explained and evaluated. Key themes include the tension between multiculturalism and gender equality, cultural relativism and the limits of liberal toleration, and the impact of multicultural policies on social cohesion ethnic conflict. Murphy also surveys the legal practices and policies enacted to accommodate multiculturalism, drawing on examples from the Americas, Australasia, Europe, Asia and the Middle East. Multiculturalism: A Critical Introduction is an ideal starting point for anyone coming to the topic for the first time as well as those already familiar with some of the key issues.
## Theory and Reality

How does science work? Does it tell us what the world is "really" like? What makes it different from other ways of understanding the universe? In Theory and Reality, Peter Godfrey-Smith addresses these questions by taking the reader on a grand tour of one hundred years of debate about science. The result is a completely accessible introduction to the main themes of the philosophy of science. Intended for undergraduates and general readers with no prior background in philosophy, Theory and Reality covers logical positivism; the problems of induction and confirmation; Karl Popper's theory of science; Thomas Kuhn and "scientific revolutions"; the views of Imre Lakatos, Larry Laudan, and Paul Feyerabend; and challenges to the field from sociology of science, feminism, and science studies. The book then looks in more detail at some specific problems and theories, including scientific realism, the theory-ladeness of observation, scientific explanation, and Bayesianism. Finally, Godfrey-Smith defends a form of philosophical naturalism as the best way to solve the main problems in the field. Throughout the text he points out connections between philosophical debates and wider discussions about science in recent decades, such as the infamous "science wars." Examples and asides engage the beginning student; a glossary of terms explains key concepts; and suggestions for further reading are included at the end of each chapter. However, this is a textbook that doesn't feel like a textbook because it captures the historical drama of changes in how science has been conceived over the last one hundred years. Like no other text in this field, Theory and Reality combines a survey of recent history of the philosophy of science with current key debates in language that any beginning scholar or critical reader can follow.
## Principia Mathematica

## Philosophy of Religion

This text combines an introduction to the themes traditionally covered in the philosophy of religion with contemporary developments in the discipline, such as natural histories of religion and feminist approaches.
## Modalities

These papers cover important themes such as extensionality, the necessity of identity, the conception of proper names as 'tags', essentialism, substitutional quantification, and possibilia and possible worlds. What emerges from them is a robust defence of quantified modal logic in the light of a host of objections, particularly from Quine.

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*A Critical Introduction*

Author: Michael Potter

Publisher: Clarendon Press

ISBN: 0191556432

Category: Philosophy

Page: 360

View: 7063

*A Critical Introduction*

Author: Michael Potter

Publisher: Clarendon Press

ISBN: 0191556432

Category: Philosophy

Page: 360

View: 1756

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*An Introduction*

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ISBN: 9780198533993

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ISBN: 9780691024479

Category: Science

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ISBN: 1134461755

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Publisher: Cambridge University Press

ISBN: 110704927X

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*An Introduction*

Author: David Bostock

Publisher: John Wiley & Sons

ISBN: 1405189924

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ISBN: 1136520104

Category: Philosophy

Page: 208

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Publisher: University of Chicago Press

ISBN: 9780226300610

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Category: Logic, Symbolic and mathematical

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ISBN: 0745638686

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Author: Ruth Barcan Marcus

Publisher: Oxford University Press on Demand

ISBN: 0195096576

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