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Anti-admissible sets

Published online by Cambridge University Press:  12 March 2014

Jacob Lurie
Jacob Lurie*
Affiliation:
6611 Braeburn Parkway, Bethesda, Maryland 20817, USA E-mail: [email protected]
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Abstract

Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the “circular logic” of [3], This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical “extension” to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU−, that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between “non-wellfounded sets over the admissible set A” and the fragment LA of the modal language L∞.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 407 - 435
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Aczel, Peter, Non-well-founded sets, CSLI Publications, Stanford, 1988.Google Scholar
[2] Barwise, Jon, Admissible sets and structures, Springer-Verlag, Berlin, 1975.Google Scholar
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[4] Barwise, Jon and Moss, Lawrence, Vicious circles, CSLI Publications, Stanford, 1996.Google Scholar
[5] Sacks, Gerald E., Higher recursion theory, Springer-Verlag, New York, 1990.Google Scholar