Paradoxes and Inconsistent Mathematics [Hardcover]

Why are there paradoxes? This book uses paraconsistent logic to develop the mathematics to find out.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.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.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.Part I. What are the Paradoxes?: Introduction to an inconsistent world; 1. Paradoxes; or, 'here in the presence of an absurdity'; Part II. How to Face the Paradoxes?: 2. In search of a uniform solution; 3. Metatheory and naive theory; 4. Prolegomena to any future inconsistent mathematics. Part III. Where are the Paradoxes?: 5. Set theory; 6. Arithmetic; 7. Algebra; 8. Real analysis; 9. Topology. Part IV. Why Are there Paradoxes?: 10. Ordinary paral³i