Quine, New Foundations, and the Philosophy of Set Theory

Description

Quine's set theory, New Foundations, has often been treated as an anomaly in the history and philosophy of set theory. In this book, Sean Morris shows that it is in fact well-motivated, emerging in a natural way from the early development of set theory. Morris introduces and explores the notion of set theory as explication: the view that there is no single correct axiomatization of set theory, but rather that the various axiomatizations all serve to explicate the notion of set and are judged largely according to pragmatic criteria. Morris also brings out the important interplay between New Foundations, Quine's philosophy of set theory, and his philosophy more generally. We see that his early technical work in logic foreshadows his later famed naturalism, with his philosophy of set theory playing a crucial role in his primary philosophical project of clarifying our conceptual scheme and specifically its logical and mathematical components. Cover Half Title Title page Imprints page Dedication Contents Preface Introduction Part I Set Theory's Beginnings Chapter 1 Cantor and the Early Development of Set Theory 1.1 The Beginnings of Set Theory as a Mathematical Discipline 1.2 From the Potential to the Actual Infinite 1.3 Cardinals, Ordinals, and the Continuum Problem Chapter 2 Cantor, Russell, and Zermelo and the Set-Theoretic Paradoxes 2.1 Russell and the Discovery of the Paradoxes 2.2 Avoiding the Paradoxes: Cantor and the Absolute Infinite 2.3 Resolving the Paradoxes: Zermelo and Russell Chapter 3 New Foundations and the Beginnings of Quine's Philosophy of Set Theory3.1 From Russell and Zermelo to New Foundations 3.2 "Contradictions Really Scare Me": New Foundations and the Paradoxes 3.3 Some Concluding Remarks on the Significance of Consistency Proofs Part II Quine, Set Theory, and Philosophy Chapter 4 Quine's Philosophy of Set Theory 4.1 Background in Russell 4.2 Early Developments: From "The Logic of Sequences" to New Foundations 4.3 New Foundations as Philosophy of Set Theory Chapter 5 Clarifying Our Conceptual Scheme: Set Theory and the Role of Explication 5.1 Clarifying Our Conceptual Scheme5.2 Explication 5.3 Quine's Mature Philosophy of Set Theory: Set Theory and Its Logic 5.4 An Addendum on Quine and Carnap on Tolerance and Set Theory Part III New Foundations and the Philosophy of Set Theory Chapter 6 The Iterative Conception and Set Theory 6.1 Boolos's Exposition 6.2 The Iterative Conception As Set Theory 6.3 The Iterative Conception and Zermelo's Axioms Chapter 7 New Foundations, the Axiom of Choice, and Arithmetic 7.1 The Axiom of Choice 7.2 Arithmetic in New Foundations 7.3 Exploring the Set-Theoretic Universe Bibliography