Nonstandard and boolean valued models, and interactions with combinatorics, dynamics, and sheaf theory

This dissertation is based on two different works. The first one pertains the model-theoretic and structural properties of boolean valued models, and their connections with sheaf theory. The second one covers some applications of nonstandard methods and ultrafilters methods in combinatorial number theory and in topological dynamics. The material is divided into three main parts. The first part, concerning the connections between the theory of boolean valued models and the theory of sheaves, aims to present the general theory of boolean valued models in a unified picture. First, we show that not all the boolean valued models are full (i.e. satisfy Łòs Theorem), and that the mixing property is strictly stronger than fullness. Then we characterize the fullness property both in algebraic terms (i.e. in terms of the semantic of the model) and in topological terms, using the notion of étalé bundle. Moreover, we expand Monro’s duality between boolean valued models and presheaves on boolean algebras. Motivated by this duality, our main result gives a presentation of the sheafification of presheaves on boolean algebras by means of sheaves of sections of suitably constructed bundles. Some consequences in the theory of boolean ultrapowers are also explored. The second part deals with some applications of nonstandard and ultrafilters methods in combinatorial number theory. This study was motivated by a recent deep result by Moreira-Richter-Robertson, who solved in the positive an old conjecture of Erdős’. Precisely, they showed that every set of natural numbers of positive Banach density contains the sum of two infinite sets. Their proof uses advanced tools of ergodic theory whose combinatorial content is not transparent, and our intent was to find a more direct, combinatorial proof that could help extend the result. We succeded in finding a purely combinatorial condition which implies the Moreira-Richter-Robertson’s Theorem, we conjecture that that condition holds for every set of positive Banach density, but we could not prove it. The other main result of this part is a generalization of a theorem of Jin’s, stating that, if a set of natural numbers has positive Banach density, then almost every point in the closure of its orbit in the Bernoulli shift has positive asymptotic density, which is the density calculated on initial intervals. We prove that the same property also holds for every density computed on any given increasing sequence of intervals, not only for initial ones. From this result we obtain several applications on the intersections of sets with positive Banach density and their shifts. The third part uses hypernatural numbers of nonstandard analysis and ultrafilters in the context of topological dynamics and measure-preserving transformations. Our goal is to present both the space of ultrafilters and the Bernoulli shift on the Cantor space as universal topological dynamics, and the hyperfinite intervals in ∗N (the nonstandard extension of the natural numbers) as universal measure-preserving dynamical systems. We show that the recurrence behavior of ultrafilters is completely determined by their dynamical effect in the Cantor space, and we use the notion of finite embeddability for discussing the kind of sets that can be found in the closure of the orbit of a fixed set of natural numbers under the Bernoulli shift. Then, we characterize the syndeticity of a set by looking at the uniformly recurrent points in its orbit, and we distinguish the recurrence behavior of sets of natural numbers from that of k-colorings of the natural numbers with k > 2. Finally, we study the measures on the space of ultrafilters βN which are induced by the Loeb counting measure on some hyperfinite interval of ∗N. We use this study to prove that every Borel probability measure on a compact metric space admitting a continuous measure-preserving transformation with generic points is actually a pushforward of the Loeb measure on some hyperfinite interval of ∗N, and we use this property to show a few facts about the measures on the Cantor space