Issues in commonsense set theory

[appeared in Artificial Intelligence Review, 1994-5]

Müjdat Pakkan and Varol Akman

Abstract:

The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given.

``... among all the mathematical theories,
it is just the theory of sets that requires
clarification more than any other.''
(Mostowski 1979)

• INTRODUCTION
• CLASSICAL SET THEORY
• Earliest Developments
• Alternate Approaches and Axiomatizations
• ZF Set Theory
• ALTERNATIVE SET THEORIES
• Admissible Set Theory
• Hyperset Theory
• Equations in the AFA Universe
• AFA and Relational Databases
• COMMONSENSE SET THEORY
• Motivation
• Desirable Properties
• Common Knowledge
• Possible Membership
• Cardinality and Well-Ordering
• Some Interesting Attempts
• Perlis's Commonsense Set Theory
• Zadrozny on Cardinalities and Well-Orderings
• Protosets of Mislove et al.
• Barwise's Situated Set Theory
• CONCLUSION
• ACKNOWLEDGMENTS
• DEDICATION
• NOTES
• REFERENCES
• About this document ...