Set Theory and its Philosophy

Author: Michael Potter
Publisher: Clarendon Press
ISBN: 9780191556432
Release Date: 2004-01-15
Genre: 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

Author: Michael Potter
Publisher: Clarendon Press
ISBN: 9780191556432
Release Date: 2004-01-15
Genre: 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

Author: Michael D. Potter
Publisher: University of Hawaii Press
ISBN: 0199270414
Release Date: 2004
Genre: Mathematics

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

Author: Michael D. Potter
Publisher:
ISBN: 0191699446
Release Date: 2004
Genre: Set theory

Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. This is a comprehensive philosophical introduction to the field offering a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates.

Handbook of Set Theory

Author: Matthew Foreman
Publisher: Springer Science & Business Media
ISBN: 9781402057649
Release Date: 2009-12-10
Genre: Mathematics

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

Author: G. Takeuti
Publisher: Springer Science & Business Media
ISBN: 9781468499155
Release Date: 2013-12-01
Genre: Mathematics

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.

Georg Cantor

Author: Joseph Warren Dauben
Publisher: Princeton University Press
ISBN: 0691024472
Release Date: 1990
Genre: Science

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.

Sets

Author: Michael D. Potter
Publisher: Oxford University Press on Demand
ISBN: 0198533993
Release Date: 1990
Genre: Ensembles, Théorie des

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.

Alain Badiou

Author: Jason Barker
Publisher: Pluto Press
ISBN: 0745318002
Release Date: 2002
Genre: Philosophy

A clear and concise introduction to the political philosophy of Alain Badiou, centred in a political context.

Introduction to Logic

Author: Patrick Suppes
Publisher: Courier Corporation
ISBN: 9780486138053
Release Date: 2012-07-12
Genre: Mathematics

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.

Theories of Democracy

Author: Frank Cunningham
Publisher: Routledge
ISBN: 9781134584956
Release Date: 2002-11-01
Genre: Philosophy

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.

Badiou and Indifferent Being

Author: William Watkin
Publisher: Bloomsbury Publishing
ISBN: 9781350015654
Release Date: 2017-09-21
Genre: Philosophy

The first critical work to attempt the mammoth undertaking of reading Badiou's Being and Event as part of a sequence has often surprising, occasionally controversial results. Looking back on its publication Badiou declared: "I had inscribed my name in the history of philosophy†?. Later he was brave enough to admit that this inscription needed correction. The central elements of Badiou's philosophy only make sense when Being and Event is read through the corrective prism of its sequel, Logics of Worlds, published nearly twenty years later. At the same time as presenting the only complete overview of Badiou's philosophical project, this book is also the first to draw out the central component of Badiou's ontology: indifference. Concentrating on its use across the core elements Being and Event-the void, the multiple, the set and the event-Watkin demonstrates that no account of Badiou's ontology is complete unless it accepts that Badiou's philosophy is primarily a presentation of indifferent being. Badiou and Indifferent Being provides a detailed and lively section by section reading of Badiou's foundational work. It is a seminal source text for all Badiou readers.

Philosophy and Educational Policy

Author: Christopher Winch
Publisher: Psychology Press
ISBN: 0415369576
Release Date: 2004
Genre: Education

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.

Rights

Author: Tom Campbell
Publisher: Routledge
ISBN: 9781134461752
Release Date: 2011-02-25
Genre: Philosophy

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.