We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations. As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from (...) both below and above. Then we show that [Formula: see text] is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided. (shrink)

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is an homeomorphism (...) between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$. (shrink)

This paper has three parts. First, we establish some of the basic model theoretic facts about, the Tsirelson space of Figiel and Johnson [20]. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model‐theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) has the non independence property (NIP); (2) every Banach space that is ℵ0‐categorical up to small perturbations embeds c0 or () almost isometrically; (...) consequently the (continuous) first‐order theory of does not characterize, up to almost isometric isomorphism. (shrink)

Given a continuous and isometric action of a Polish group G on an adequate Polish topometric space $(X,\tau,\rho )$ and $x \in X$, we find a necessary and sufficient condition for $\overline {Gx}^{\rho }$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.

We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. In order to do this,•We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise א0-categorical stable theories in which the last two agree.•We characterise sequences that admit almost indiscernible sub-sequences.•We apply these tools to the theory of (...) atomless random variables (ARV). We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes and Rosenthal. (shrink)