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Non-Well-Founded Sets (left for the Study of Language and Information - Lecture Notes): Peter Aczel

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 modeling 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.

Table of Contents

Foreword

Preface

Introduction

Part One The Anti-Foundation Axiom

1 Introducing the Axiom

2 The Axiom in More Detail

3 A Model of the Axiom

Part Two Variants of the Anti-Foundation Axiom

4 Variants Using a Regular Bisimulation

5 Another Variant

Part Three On Using the Anti-Foundation Axiom

6 Fixed Points of Set Continuous Operators

7 The Special Final Coalgebra Theorem

8 An Application to Communicating Systems

Appendices

A. Notes Towards a History

B. Background Set Theory

References

Index of Definitions

Index of Named Axioms and Results

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