Non-well-founded structures arise in a variety of ways in the semantics of both natural and formal languages. Two examples are non-well-founded situations and non-terminating computational processes. A natural modelling of such structures in set theory requires the use of non-well-founded sets. This text presents the mathematical background to the anti-foundation axiom and related axioms that imply the existence of non-well-founded sets when used in place of the axiom of foundation in axiomatic set theory.

These essays evolved from research presented at the Third International Conference on situation theory and its applications. Situation Theory is the result of an interdisciplinary effort to create a full-fledged theory of information. Created by scholars and scientists from cognitive science, computer science and AI, linguistics, logic, philosophy, and mathematics, it aims to provide a common set of tools for the analysis of phenomena from all of these fields. The research presented in this volume reflects a growing international and interdisciplinary activity of importance to many fields concerned with the information. Peter Aczel is professor of mathematical logic and computer logic at Manchester University. David Israel is a senoir computer scientist in the Artificial Intelligence Center at SRI International abd a consulting professor in the Philosophy Department at Stanford University. Yasuhiro Katagiri is a research scientist in the Information Science Research Laboratory of NTT Basic Research Laboratories. Stanley Peters is professor of linguistics and symbolic systems at Stanford University.