Topological set theories and hyperuniverses

We give a new set theoretic system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK and the theory of hyperuniverses. On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF. We single out the additional axiom of the universal set as the one that increases the consistency strength to that of GPK and explore several other axioms and interrelations between those theories. Hyperuniverses are a natural class of models for theories with a universal set. The Aleph_0- and Aleph_1-dimensional Cantor cubes are examples of hyperuniverses with additivity Aleph_0, because they are homeomorphic to their hyperspace. We prove that in the realm of spaces with uncountable additivity, none of the generalized Cantor cubes has that property. Finally, we give two complementary constructions of hyperuniverses which generalize many of the constructions found in the literature and produce initial and terminal hyperuniverses

Wellordering proofs for metapredicative Mahlo

In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed shar

A fixed-point theorem in a category of compact metric spaces

AbstractVarious results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question.Following a new kind of approach, based on the notion of η-isometry, we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embedding-projection pairs

Event structures for Petri nets with persistence

Event structures are a well-accepted model of concurrency. In a seminal paper by Nielsen, Plotkin and Winskel, they are used to establish a bridge between the theory of domains and the approach to concurrency proposed by Petri. A basic role is played by an unfolding construction that maps (safe) Petri nets into a subclass of event structures, called prime event structures, where each event has a uniquely determined set of causes. Prime event structures, in turn, can be identified with their domain of configurations. At a categorical level, this is nicely formalised by Winskel as a chain of coreflections. Contrary to prime event structures, general event structures allow for the presence of disjunctive causes, i.e., events can be enabled by distinct minimal sets of events. In this paper, we extend the connection between Petri nets and event structures in order to include disjunctive causes. In particular, we show that, at the level of nets, disjunctive causes are well accounted for by persistent places. These are places where tokens, once generated, can be used several times without being consumed and where multiple tokens are interpreted collectively, i.e., their histories are inessential. Generalising the work on ordinary nets, Petri nets with persistence are related to a new subclass of general event structures, called locally connected, by means of a chain of coreflections relying on an unfolding construction