The positive comprehension principle in set theory
by
Olivier Esser
Universite Libre de Bruxelles

The positive theory GPK⁺_\infty, due to M. Forti, R. Hinnion, F. Honsell, E. Weydert, M. Boffa (see [5]) has a comprehension scheme for bounded positive formulas and an axiom scheme of closure: for any class A of sets (i.e. any definable collection of sets), there is a smallest set [`A] including A. This closure's operation behaves like a topological closure. This theory has a universal set V={x | x=x} (``x=x'' is positive).

This theory is mutually interpretable with KM + ``On has the tree-property''; KM is the Kelley-Morse class-theory; ``On has the tree-property'' is the natural translation to the class of ordinals of the corresponding notion for cardinals in ZF ([2]). An interesting result on GPK⁺_\infty is that the axiom of choice is inconsistent with it ([3]).

Natural models of this theory are hyperuniverses. These can be obtained by taking the Cauchy completion of some suitable first order structures. We need some large cardinals properties (mildly ineffable cardinals, see e.g. [1]) in order to get the Cauchy completion \kappa-compact; which is here an essential property ([4]).

References

[1] Di Prisco C. A. and Zwicker S. Flipping properties and supercompact cardinals, Fundamenta Mathematicae CIX (1980), pp. 31-36.

[2] Esser O. On the Consistency of a Positive Theory. Mathematical Logic Quarterly 45 (1999).

[3] Esser O. Inconsistency of the axiom of choice with GPK. To appear in J. Symb. Logic.

[4] Esser O. Mildly ineffable cardinals and hyperuniverses, submitted.

[5] Forti M. and Hinnion R. The Consistency Problem for Positive Comprehension Principles. J. Symb. Logic 54 (1989), pp. 1401-1418.

http://homepages.ulb.ac.be/~oesser

Date received: February 7, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagb-04.