Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena (CSLI Lecture Notes) by Jon Barwise (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review ‘ … a book to learn from.’ L’Enseignement Mathématique Book Description Many assume that circular phenomena and mathematical rigour are irreconcilable. Barwise and Moss have undertaken to prove this assumption false. Vicious Circles is intended for use by researchers who use hypersets, although the book is accessible to people with widely differing backgrounds and interests. About the Author Lawrence S. Moss is professor of mathematics; director of the Program in Pure and Applied Logic; an adjunct professor of computer science, informatics, linguistics, and philosophy; and a member of the Programs in Cognitive Science and Computational Linguistics, all at Indiana University, Bloomington. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐I have a degree in mathematics from Cambridge University and I’m interested in Foundations and paradoxes, such as the two-player game, G, where the first player picks any finite game, then they alternate moves, with the second player playing first. Is the game finite or not?Yes, because it’s a finite game with just one extra move. No, because if it’s finite, the first player could choose G, whereupon the second player would make the first move, which could be to choose G as the finite game to play, …Well, this is tackled fairly late in the book, long after I got lost.Some of the unfamiliarity stems from the author’s abandonment of the axiom of foundation, which prohibits infinitely nested sets, such as {{{…{}….}}}, but I was willing to accept this for the sake of learning something new. Still, even the early exercises left me behind. Perhaps it’s just too long since I was a college student.

⭐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.

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