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.
was professor of philosophy, mathematics, and computer science at Indiana University and one of the founding members of CSLI. is Associate Professor of Mathematics at Indiana University.
- Part I Background
- 1 Introduction
- 2 Background on set theory
- Part II Vicious Circles
- 3 Circularity in computer science
- 4 Circularity in philosophy
- 5 Circularity and paradox
- Part III Basic Theory
- 6 Circularity in philosophy
- 7 Circularity and paradox
- Part IV Elementary applications
- 8 Graphs
- 9 Modal logic
- 10 Streams
- 11 Games
- 12 Modeling the semantic paradoxes
- Part V Further Theory
- 13 Greatest fixed points
- 14 Uniform operators
- 15 Corecursion
- Part VI Further Applications
- 17 Some applications
- 17 TModeling partial information
- 18 Circularity and the notion of set
- 19 Conclusions and future directions