We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of (...) the set instead of the cardinality, is ${\aleph_0}$ . In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs. (shrink)

This paper is a general introduction to Positive Logic, where only what we call h-inductive sentences are under consideration, allowing the extension to homomorphisms of model-theoric notions which are classically associated to embeddings; in particular, the existentially closed models, that were primitively defined by Abraham Robinson, become here positively closed models. It accounts for recent results in this domain, and is oriented towards the positivisation of Jonsson theories.

In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of 0-stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over 0/, Löwenheim–Skolem number being ω, -tameness and a property we call finite character. We also (...) start the studies of these classes from the 0-stable case. Stability in 0 and -tameness can be replaced by categoricity above the Hanf number. Finite character is the main novelty of this paper. Almost all examples of AEC’s have this property and it allows us to use weak types, as we call them, in place of Galois types. (shrink)

We investigate properties of accessible categories with directed colimits and their relationship with categories arising from ShelahʼsElementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.