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1. Isomorphic spectrum and isomorphic length of a Banach space

   https://doi.org/10.15330/cmp.12.1.88-93  

   Fotiy, O.; Ostrovskii, M.; Popov, M. (June 2020, Carpathian Mathematical Publications)

   null (Ed.)

   We prove that, given any ordinal $$\delta < \omega_2$$, there exists a transfinite $$\delta$$-sequence of separable Banach spaces $$(X_\alpha)_{\alpha < \delta}$$ such that $$X_\alpha$$ embeds isomorphically into $$X_\beta$$ and contains no subspace isomorphic to $$X_\beta$$ for all $$\alpha < \beta < \delta$$. All these spaces are subspaces of the Banach space $$E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$$, where $$1 \leq p < 2$$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $$\delta$$ of continuum cardinality. 

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2. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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3. COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

   https://doi.org/10.1017/jsl.2019.28  

   GUNATILLEKA, DANUL K. (September 2019, The Journal of Symbolic Logic)

   Abstract We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $$\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $$\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $$ if and only if the dimension of $$\mathfrak{M}_\beta $$ is at most the dimension of $$\mathfrak{M}_\gamma $$ and that each countable model is isomorphic to some $$\mathfrak{M}_\beta $$ . We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω -stable theory with a nonlocally modular regular type, answering a question of Pillay in [11]. 

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4. Limit models in strictly stable abstract elementary classes

   https://doi.org/10.1002/malq.202200075  

   Boney, Will; VanDieren, Monica_M (December 2024, Mathematical Logic Quarterly)

   Abstract In this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfyingthe joint embedding and amalgamation properties with no maximal model of cardinality ,stability in ,,continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed. 

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5. COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION

   https://doi.org/10.1017/jsl.2022.78  

   CHAN, WILLIAM; JACKSON, STEPHEN; TRANG, NAM (November 2022, The Journal of Symbolic Logic)

   Abstract Assume $$\mathsf {ZF} + \mathsf {AD}$$ and all sets of reals are Suslin. Let $$\Gamma $$ be a pointclass closed under $$\wedge $$ , $$\vee $$ , $$\forall ^{\mathbb {R}}$$ , continuous substitution, and has the scale property. Let $$\kappa = \delta (\Gamma )$$ be the supremum of the length of prewellorderings on $$\mathbb {R}$$ which belong to $$\Delta = \Gamma \cap \check \Gamma $$ . Let $$\mathsf {club}$$ denote the collection of club subsets of $$\kappa $$ . Then the countable length everywhere club uniformization holds for $$\kappa $$ : For every relation $$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$$ with the property that for all $$\ell \in {}^{<{\omega _1}}\kappa $$ and clubs $$C \subseteq D \subseteq \kappa $$ , $$R(\ell ,D)$$ implies $$R(\ell ,C)$$ , there is a uniformization function $$\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$$ with the property that for all $$\ell \in \mathrm {dom}(R)$$ , $$R(\ell ,\Lambda (\ell ))$$ . In particular, under these assumptions, for all $$n \in \omega $$ , $$\boldsymbol {\delta }^1_{2n + 1}$$ satisfies the countable length everywhere club uniformization. 

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