Author: James Ernest HansonPublisher:
ISBN:
Category :
Languages : en
Pages : 588
Book Description
After a self-contained development of continuous first-order logic, we study the phenomena of definability and categoricity in continuous logic. The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. We investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end, we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out, and we show that [omega]-stable theories and randomizations of arbitrary continuous theories are dictionaric. We also introduce, in the context of Banach theories, 'indiscernible subspaces,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable outside of the context of inseparably categorical theories. We construct or present a slew of counterexamples, including an [omega]-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension. In order to investigate notions of approximate categoricity, we give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of Ben Yaacov and Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. We introduce distortion systems, which are a mild generalization of perturbation systems. With this we explicitly exhibit Scott sentences for perturbation systems, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically, by distortion systems, and semantically, by certain elementary classes of two-sorted structures that witness approximate isomorphism. We also make progress towards an analog of Morley's theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal [kappa] such that every model of size [kappa] is 'approximately saturated,' in the appropriate sense, then the same is true for all uncountable cardinalities. Finally, we present some non-trivial examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.
