Universal Set Theory and Strong Modal Reflection Principles

Alexandru Baltag

I consider infinitary modal logic L_{∞}^{♢} as a foundational language for set theory. The Barwise-Moss semantics for modal logic (see [4]) is given by considering sets and classes as Kripke structures, with converse-membership as the accessibility relation. I use this semantics to axiomatize and model a nonclassical, non-wellfounded universe of sets, containing a “universal” set U and other “large” sets. This universe can be seen as the largest extension of Peter Aczel’s non-wellfounded universe which still satisfies Aczel’s Strong Extensionality principle. My central axiom SAF A (“Super-Antifoundation Axiom”) is a strengthening of Aczel’s AF A, stated as a strong Reflection Principle L_{∞}^{♢}. My consistency proof for this set theory uses a generalization of the standard “canonical model” construction for modal logic. The resulting model is isomorphic to one of the models constructed by researchers on “topological set theory” ([5], [6] etc). I use this set theory to give non-classical solutions to classical set-theoretical paradoxes, to prove fixed-point theorems relating recursion and corecursion, to formalize a “self-referential” model theory and a “reflexive” category theory, to provide natural solutions for domain equations, and to justify some well-known “large cardinal” axioms.

1 A. Baltag, STS: A Structural Theory of Sets, Ph. D. Dissertation, Indiana University 1998.

2 A. Baltag, A Structural Theory of Sets, Proceeedings of “Advances in Modal Logic” (AiML’98), 32-41. Uppsala, Sweden, 1998.

3 A. Baltag, STS: A Structural Theory of Sets, Logic Journal of the IGPL, vol. 7:4, 481-515, Oxford Univ. Press, July 1999

4 J. Barwise, L.S. Moss, Vicious Circles, CSLI Lecture Notes no. 60, Stanford 1996.

5 M. Forti, F. Honsell The Consistency Problem for Positive Comprehension Principles, Journal of Symbolic Logic 54 (4):1401-1417, 1989.

6 M. Forti, F. Honsell A General Construction of Hyperuniverses, Quadermi dell’ Istituto di Matematiche Applicate “U. Dini”, no. 9, 1992.

Address: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

E-mail: abaltag@cwi.nl