Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena

by Lawrence S. Moss, Lawrence Moss, Jon Barwise

Book Overview

Circular analyses of philosophical, linguistic, or computational phenomena have been attacked on the assumption that they conflict with mathematical rigour. Barwise and Moss have undertaken to prove... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 1575860090
• ISBN13: 9781575860091
• Release Date: June 1996
• Publisher: Center for the Study of Language and Informat
• Length: 400 Pages
• Weight: 1.60 lbs.
• Dimensions: 1.4" x 6.3" x 9.3"

Customer Reviews

First systematic exposition of the Anti-Foundation Axiom

As the title suggests, this is the first systematic exposition of classic set theory without the axiom of foundation. What replaces it, the anti-foundation axiom, allows sets to be members of themselves and it is this type of circularity that, as the authors claim, lies at the heart of understanding knowledge in interacting systems (like computing machines or game-theoretic agents or Liar-type sentences that refer to themselves). What makes the whole endeavour work is that this new axiom is still consistent with the rest of ZF theory (a fact that is proved in the book) and in this sense the new theory can be thought as an "extention" of the traditional hierarchical construction of sets. The book is written in textbook style in that it presents the material methodically and it is reasonably self-contained (a basic understanding of set theory and adequate motivation are enough). I would have given 4 stars for poor binding and some typos (nothing serious though), but the quality of presentation and the fact that it includes answers to all problems more than make up for it.

Fasciniating look at a new extension of set theory

This book discusses recent advances in the general field of set theory. The authors study a variant of ZF in which the axiom of foundation is replaced by a new axiom allowing non-well-founded sets. Just as the naturals can be extended to the integers, and the integers to the rationals, and the reals to the complex numbers, in each case by positing new numbers that are the solutions to a class of equations, so this book posits an extension to any model of set theory consisting of the solutions to a class of (systems of) equations having no solutions in ZF. The simplest example is the equation x = {x}, whose solution, x = {{{{...}}}} (infinitely deep)is not permitted in ZF, but exists and is unique in the authors' theory.The purpose of this extension to ZF is to create a set theory in which certain circular or infinite phenomena from computer science and other fields, e.g. cyclic data streams, can be much more directly modeled than is now possible in ZF. Currently in ZF in order to represent a cyclic data stream one has to develop the aparatus for natural numbers, and then represent the stream to be a function from the natural numbers into some suitable set representing the type of data. But in the author's set theory the stream could be represented as an unfounded set that is the solution to a simple equation, and many of its properties could then be more easily deduced without resort to arithmetic.I found this book absolutely fascinating, and I highly recommend it to anyone who has had a course in set theory. The theory in the book is quite elegant and satisfying. I was delighted to learn that there is still room for new variations of the axioms of set theory, a subject I thought (probably naively) had been fairly static for 60 years.