On the Generalized Positive Comprehension Principle

Olivier Esser

In this communication, we present the theory GPK_{∞}^{+} (due to M. Forti, R. Hinnion, F. Honsell, E. Weydert, M. Boffa). This theory 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 \overline{A} including A.

We can show that this closure operation behaves like a topological closure. It is thus natural to call a class A closed iff \overline{A} = A, which is equivalent to say that A is a set. Notice that this theory has a universal set V = {x | x = x} (“x = x” is positive).

Our main result about this theory is that it is mutually interpretable with KM + “On has the treeproperty”; KM is the Kelley-Morse class-theory; “On has the tree-proprety” is the natural translation to the class of ordinals of the corresponding notion for cardinals in ZF. The sets of the interpretation of KM in GPK_{∞}^{+} are the well-founded sets of GPK_{∞}^{+} and the classes are the classes (in the sense of GPK_{∞}^{+}) of well-founded sets; these classes can be represented in GPK_{∞}^{+} by their closure using the fact that the well-founded sets are isolated in the topological sense. The proof of this result is among other inspired from results on hyperuniverses, the “natural” models of GPK_{∞}^{+} (see f.e. [3]).

Another interesting result on GPK_{∞}^{+} is that the axiom of choice is inconsistent with it ([2]).

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

[2] Esser O. Inconsistency of the axiom of choice with GPK_{∞}^{+}. To appear in J. Symb. Logic.

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

(For a complete list of references, see [2])

Address: Université Libre de Bruxelles, Campus de la Plaine, Service de Logique C.P. 211, Boulevard du Triomphe, B1050 Bruxelles (Belgium)

E-mail: oesser@ulb.ac.be