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.

Peter Aczel is reader in mathematical logic at Manchester University

- 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-Formation 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 Axioms and Results

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