It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a wellknown set theory based on an extended positive comprehension scheme.

Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.

This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory” (the universes discussed here concern, roughly speaking : stratified sets, partial sets, positive sets, paradoxical sets and double sets).

In this paper, we study when a set-continuous operator has a fixed-point that is the intersection of a directed family. The framework of our study is the Kelley-Morse theory KMC– and the Gödel-Bernays theory GBC–, both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC– that cannot be proved in GBC–. On the other hand, we study conditions on directed superclasses in GBC– in order (...) that their intersection is a fixed-point of a set-continuous operator. Finally, we illustrate our results with a solution to the liar paradox and a construction of maximal bisimulations. (shrink)

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

The tree-property and its variants make sense also for directed sets and even for partially ordered sets. A combinatoria approach is developed here, with characterizations and criteria involving adequate families of special substructures of directed sets. These substructures form a natural hierarchy that is also investigated.

This is a study of the set of the Malitz-completions of a given infinite first-order structure, put in relation with properties of directed sets.

The notion of “ramifiability” , usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.

We state the consistency problem of extensional partial set theory and prove two complementary results toward a definitive solution. The proof of one of our results makes use of an extension of the topological construction that was originally applied in the paraconsistent case.

The role of foundation with respect to transitive closure in the Zermelo system Z has been investigated by Boffa; our aim is to explore the role of antifoundation. We start by showing the consistency of "Z antifoundation transitive closure" relative to Z (by a technique well known for ZF). Further, we introduce a "weak replacement principle" (deductible from antifoundation and transitive closure) and study the relations among these three statements in Z via interpretations. Finally, we give some adaptations for ZF (...) without infinity. (shrink)