Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena

by Jon Barwise, Lawrence S. Moss

The subject of non-wellfounded sets came to prominence with the 1988 publication of Peter Aczel's book on the subject. Since then, a number of researchers in widely differing fields have used non-wellfounded sets (also called "hypersets") in modeling many types of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and programming languages. Vicious Circles offers an introduction to this fascinating and timely topic. Written as a book to learn from, theoretical points are always illustrated by examples from the applications and by exercises whose solutions are also presented. The text is suitable for use in a classroom, seminar, or for individual study. In addition to presenting the basic material on hypersets and their applications, this volume thoroughly develops the mathematics behind solving systems of set equations, greatest fixed points, coinduction, and corecursion. Much of this material has not appeared before. The application chapters also contain new material on modal logic and new explorations of paradoxes from semantics and game theory.

Genres: Philosophy, Mathematics, Logic

390 pages, Paperback

First published June 1, 1996

Reviews

[Lexi Pasi · Rating 5 out of 5]

Admittedly, I haven't yet read this book cover-to-cover and don't even own my own copy, as I was merely borrowing the copy from the MSRI library during a brief visit there. But wow, what a book! This work strikes a great balance between speculation and formalization, and shines a light on perspectives on self-reference in philosophy, computer science, linguistics, and other areas. It's surprisingly accessible for the level of rigor it achieves, and while some familiarity with set-theory is a helpful pre-requisite, I imagine that non-mathematicians might also be able to enjoy this book with a bit of patience and some willingness to cross-reference unfamiliar topics.

[Axel Barceló · Rating 4 out of 5]

I cannot believe I was lucky enough to attend Moss and Barwise's classes while they were preparing this awesome classic