Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.
Author(s): Zach Weber
Edition: 1
Publisher: Cambridge University Press
Year: 2021
Language: English
Pages: 260
Tags: Logical Paradoxes; Dialetheic Paraconsistency;
Cover
Half-title
Title page
Copyright information
Dedication
Contents
Preface
Part I What Are the Paradoxes?
Introduction to an Inconsistent World
0.1 The Problem
0.2 The Choices
0.3 Prospectus: Fixed Points
1 Paradoxes; or, ``Here in the Presence of an Absurdity''
1.1 Sets
1.2 Vagueness
1.3 Boundaries
1.4 Conclusion
Part II How to Face the Paradoxes?
2 In Search of a Uniform Solution
2.1 In Search of an Explanation
2.2 Two Schemas
2.3 Stepping Back from the Limits of Thought
3 Metatheory and Naive Theory
3.1 The Myth of Metatheory
3.2 Classical Recapture
3.3 Naive Theory
4 Prolegomena to Any Future Inconsistent Mathematics
4.1 Curry’s Paradox
4.2 Gris̆in’s Paradox and Identity
4.3 Logic
Appendix: BCK and DKQ
Part III Where Are the Paradoxes?
5 Set Theory
5.1 Elements
5.2 A Sketch of the Universe
5.3 Order
Excursus: Partitions, Equivalence Classes, and Cardinality
6 Arithmetic
6.1 Thither Paraconsistent Arithmetic!
6.2 Addition, Multiplication, and Order
Excursus: Number Theory
6.3 Descent: Inconsistency and Irrationality
7 Algebra
7.1 Algebra for Inconsistent Mathematics: A Triviality Problem
7.2 Vectors
7.3 Groups, Rings, and Fields
7.4 A Short Conclusion to a Short Chapter
8 Real Analysis
8.1 Into the Labyrinth: Real Numbers
8.2 Dedekind Cuts
8.3 Continuity; or, ``Amongst the Ghosts of Departed Quantities''
8.4 Out of the Labyrinth: The Topology of a Point
9 Topology
9.1 Closure Spaces
Excursus: Consequence as Closure
9.2 Boundaries and Connected Space
9.3 Continuity
Part IV Why Are There Paradoxes?
10 Ordinary Paradox
10.1 Dividing the Universe
10.2 The Last Horizon
10.3 A Fixed Point Where None Can Be
Bibliography
Index
