Contents
Part I Background
1. Introduction
1.1 Set theory and circularity
1.2 Preview
2. Background on set theory
2.1 Some basic operations on sets
2.2 Sets and classes
2.3 Ordinals
2.4 The Axiom of Plenitude
2.5 The Axiom of Foundation
2.6 The axioms of set theory
Part II Vicious Circles
3. Circularity in computer science
3.1 Streams
3.2 Labeled transition systems
3.3 Closures
3.4 Self-applicative programs
3.5 Common themes
4. Circularity in philosophy
4.1 Common knowledge and the Conway Paradox
4.2 Other intentional phenomena
4.3 Back to basics
4.4 Examples from other fields
5. Circularity and paradox
5.1 The liar paradox
5.2 Paradox of denotation
5.3 The hypergame paradox
5.4 Russell's paradox
5.5 Lessons from the paradoxes
Part III Basic Theory
6. The solution lemma
6.1 Modeling equations and their solutions
6.2 The solution lemma formulation of AFA
6.3 An extension of the Flat Solution Lemma
7. Bisimulation
7.1 Bisimilar systems of equations
7.2 Strong extensionality of sets
7.3 Applications of bisimulation
7.4 Computing bisimulation
8. Substitution
8.1 General systems of equations
8.2 Substitution
8.3 The general forms of the solution lemma
8.4 The algebra of substitutions
9. Building a model of ZFA
9.1 The Model
9.2 Bisimulation systems
9.3 Verifying ZFC~
94 Verifying AFA
Part IV Elementary Applications
10. Graphs
10.1 Graphs and sets they picture
10.2 Labeled graphs
10.3 Bisimilar graphs
11. Modal Logic
11.1 An introduction to modal logic
11.2 Characterizing sets by sentences
11.3 Baltag's theorems
11.4 Proof theory and completeness
11.5 Characterizing classes by modal theories
12. Games
12.1 Modeling games
12.2 Applications of games
12.3 The hypergame paradox resolved
13. The semantical paradoxes
13.1 Partial model theory
13.2 Accessible models
13.3 Truth and paradox
13.4 The liar
13.5 Reference and paradox
14. Streams
14.1 The set^∞ of streams as a fixed point
14.2 Streams, coinduction and corecursion
14.3 Stream systems
Part V Further Theory
15. Greatest fixed points
15.1 Fixed points of monotone operators
15.2 Least fixed points
15.3 Greatest fixed points
15.4 Games and fixed points
16. Uniform operators
16.1 Systems of equations as coalgebras
16.2 Morphisms
16.3 Solving coalgebras
16.4 Representing the greatest fixed points
16.5 The Solution Lemma Lemma
16.6 Allowing operations in equations
17. Corecursion
17.1 Smooth operators
17.2 The corecursion theorem
17.3 Simultaneous corecursion
17.4 Bisimulation generalized
Part VI Further Applications
18. Some important greatest fixed points
18.1 Hereditarily finite sets
18.2 Infinite binary trees
18.3 Canonical labeled transition systems
18.4 Deterministic automata and languages
18.5 Labeledsets
19. Modal logics from operators
19.1 Some example logics
19.2 Operator logics definced
19.3Characterization theorems
20. Wanted: a strongly extensional theory of classes
20.1 Paradise lost
20.2 What are ZFC adn ZFA axiomatizations of?
20.3 Four criteria
20.4 Classes as façon de pqrler
20.5 The theory of SEC0
20.6 Parting thoughts on the paradoxes
21. Past, present and future
21.1 The past
21.2 The present
21.3 The future
Appendix: definitions and results on operators
Answers to the Exercises
Bibliography
Index
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