In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) is ramifiable” implies the existence of a proper class of inaccessible cardinals. (shrink)

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK∞+. We show that in these conditions, we can interprete the Kelley‐Morse theory (KM) in GPK∞+ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo‐Fraenkel set theory (ZF) in GPK∞+. We also see what happens in (...) the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections. (shrink)

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X1 introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X1 in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It (...) includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. (shrink)

This paper is concerned with topological set theory, and particularly with Skala's and Manakos' systems for which we give a topological characterization of the models. This enables us to answer natural questions about those theories, reviewing previous results and proving new ones. One of these shows that Skala's set theory is in a sense compatible with any ‘normal’ set theory, and another appears on the semantic side as a ‘Cantor theorem’ for the category of Alexandroff spaces.

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 define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom.

This is a study of the relative interpretability of the axiom of extensionality in the positive set theory. This work has to be considered in the line of works of R. O. Gandy, D. Scott and R. Hinnion who have studied the relative interpretability of the axiom of extensionality in set theories of Zermelo and Zermelo-Fraenkel.

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.

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.

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)

The purpose of this paper is mainly to give a model of paraconsistent logic satisfying the "Frege comprehension scheme" in which we can develop standard set theory (and even much more as we shall see). This is the continuation of the work of Hinnion and Libert.

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory $GPK^+ \infty$.