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  1.  7
    Tree-like constructions in topology and modal logic.G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan & J. Van Mill - 2021 - Archive for Mathematical Logic 60 (3):265-299.
    Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.
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  2.  5
    Small sets in Mann pairs.Pantelis E. Eleftheriou - 2021 - Archive for Mathematical Logic 60 (3):317-327.
    Let \ be an expansion of a real closed field \ by a dense subgroup G of \ with the Mann property. We prove that the induced structure on G by \ eliminates imaginaries. As a consequence, every small set X definable in \ can be definably embedded into some \, uniformly in parameters. These results are proved in a more general setting, where \ is an expansion of an o-minimal structure \ by a dense set \, satisfying three tameness (...)
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  3.  6
    Quantum logic is undecidable.Tobias Fritz - 2021 - Archive for Mathematical Logic 60 (3):329-341.
    We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature \\), where ‘\’ is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent result of Slofstra in combinatorial group theory. It follows upon reinterpreting that result in terms of the hypergraph approach to quantum contextuality, for which it (...)
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  4.  8
    Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum.Sakaé Fuchino, André Ottenbreit Maschio Rodrigues & Hiroshi Sakai - 2021 - Archive for Mathematical Logic 60 (3):495-523.
    Continuing, we study the Strong Downward Löwenheim–Skolem Theorems of the stationary logic and their variations. In Fuchino et al. it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters \. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size \. We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation (...)
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  5.  6
    Another method for constructing models of not approachability and not SCH.Moti Gitik - 2021 - Archive for Mathematical Logic 60 (3):469-475.
    We present a new method of constructing a model of \SCH+\AP.
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  6.  8
    Cichoń’s diagram and localisation cardinals.Martin Goldstern & Lukas Daniel Klausner - 2021 - Archive for Mathematical Logic 60 (3):343-411.
    We reimplement the creature forcing construction used by Fischer et al. :1045–1103, 2017. https://doi.org/10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics, sometimes referred to as localisation cardinals.
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  7.  4
    Sofic Profiles of $$S(\Omega )$$ and Computability.Aleksander Ivanov - 2021 - Archive for Mathematical Logic 60 (3-4):477-494.
    We show that for every sofic chunk E there is a bijective homomorphism \, where \ is a chunk of the group of computable permutations of \ so that the approximating morphisms of E can be viewed as restrictions of permutations of \ to finite subsets of \. Using this we study some relevant effectivity conditions associated with sofic chunks and their profiles.
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  8.  6
    Keisler’s order via Boolean ultrapowers.Francesco Parente - 2021 - Archive for Mathematical Logic 60 (3):425-439.
    In this paper, we provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of ‘separation of variables’ recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in 1974.
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  9.  2
    Axiomatic theory of betweenness.Sanaz Azimipour & Pavel Naumov - 2021 - Archive for Mathematical Logic 60 (1):227-239.
    Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three sets of points. The main technical result is a sound and complete logical system describing universal properties of this relation between sets of vertices of a graph.
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  10.  3
    Selection Properties of the Split Interval and the Continuum Hypothesis.Taras Banakh - 2021 - Archive for Mathematical Logic 60 (1-2):121-133.
    We prove that every usco multimap $$\varPhi :X\rightarrow Y$$ Φ : X → Y from a metrizable separable space X to a GO-space Y has an $$F_\sigma $$ F σ -measurable selection. On the other hand, for the split interval $${\ddot{\mathbb I}}$$ I ¨ and the projection $$P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2$$ P : I ¨ 2 → I 2 of its square onto the unit square $$\mathbb I^2$$ I 2, the usco multimap $${P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb I}}}^2}$$ P - 1 : (...)
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  11.  2
    Some Complete $$\Omega $$-Powers of a One-Counter Language, for Any Borel Class of Finite Rank.Olivier Finkel & Dominique Lecomte - 2021 - Archive for Mathematical Logic 60 (1-2):161-187.
    We prove that, for any natural number \, we can find a finite alphabet \ and a finitary language L over \ accepted by a one-counter automaton, such that the \-power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$is \-complete. We prove a similar result for the class \.
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  12.  4
    Strong Downward Löwenheim–Skolem Theorems for Stationary Logics, I.Sakaé Fuchino, André Ottenbreit Maschio Rodrigues & Hiroshi Sakai - 2021 - Archive for Mathematical Logic 60 (1-2):17-47.
    This note concerns the model theoretic properties of logics extending the first-order logic with monadic second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem down to <ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<\aleph _2$$\end{document} for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into four principles. The strongest of these four is shown to be equivalent to the conjunction of CH and the (...)
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  13.  3
    Characterising Brouwer’s continuity by bar recursion on moduli of continuity.Makoto Fujiwara & Tatsuji Kawai - 2021 - Archive for Mathematical Logic 60 (1):241-263.
    We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \ to the natural numbers \ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar (...)
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  14.  5
    On countably saturated linear orders and certain class of countably saturated graphs.Ziemowit Kostana - 2021 - Archive for Mathematical Logic 60 (1):189-209.
    The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality \. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality \, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From (...)
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  15.  2
    First-order concatenation theory with bounded quantifiers.Lars Kristiansen & Juvenal Murwanashyaka - 2021 - Archive for Mathematical Logic 60 (1):77-104.
    We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.
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  16.  4
    Some Complete Ω\Documentclass[12pt]{Minimal} \Usepackage{Amsmath} \Usepackage{Wasysym} \Usepackage{Amsfonts} \Usepackage{Amssymb} \Usepackage{Amsbsy} \Usepackage{Mathrsfs} \Usepackage{Upgreek} \Setlength{\Oddsidemargin}{-69pt} \Begin{Document}$$\Omega $$\End{Document}-Powers of a One-Counter Language, for Any Borel Class of Finite Rank. [REVIEW]Dominique Lecomte & Olivier Finkel - 2021 - Archive for Mathematical Logic 60 (1-2):161-187.
    We prove that, for any natural number n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}, we can find a finite alphabet Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and a finitary language L over Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} accepted by a one-counter automaton, such that the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-power L∞:={w0w1…∈Σω∣∀i∈ωwi∈L}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (...)
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  17. Continuous Logic and Embeddings of Lebesgue Spaces.Timothy H. McNicholl - 2021 - Archive for Mathematical Logic 60 (1-2):105-119.
    We use the compactness theorem of continuous logic to give a new proof that Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^r$$\end{document} isometrically embeds into Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} whenever 1≤p≤r≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le p \le r \le 2$$\end{document}. We will also give a proof for the complex case. This will involve a new characterization of complex Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...)
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  18.  1
    A note on uniform density in weak arithmetical theories.Duccio Pianigiani & Andrea Sorbi - 2021 - Archive for Mathematical Logic 60 (1):211-225.
    Answering a question raised by Shavrukov and Visser :569–582, 2014), we show that the lattice of \-sentences ) over any computable enumerable consistent extension T of \ is uniformly dense. We also show that for every \ and \ refer to the known hierarchies of arithmetical formulas introduced by Burr for intuitionistic arithmetic) the lattices of \-sentences over any c.e. consistent extension T of the intuitionistic version of Robinson Arithmetic \ are uniformly dense. As an immediate consequence of the proof, (...)
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  19. $$AD_{Mathbb {R}}$$ A D R Implies That All Sets of Reals Are $$Theta $$ Θ Universally Baire. [REVIEW]Grigor Sargsyan - 2021 - Archive for Mathematical Logic 60 (1-2):1-15.
    We show that assuming the determinacy of all games on reals, every set of reals is Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} universally baire.
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  20.  10
    $$AD_{Mathbb {R}}$$ A D R Implies That All Sets of Reals Are $$Theta $$ Θ Universally Baire.Grigor Sargsyan - 2021 - Archive for Mathematical Logic 60 (1-2):1-15.
    We show that assuming the determinacy of all games on reals, every set of reals is \ universally baire.
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  21.  5
    Strong cell decomposition property in o-minimal traces.Somayyeh Tari - 2021 - Archive for Mathematical Logic 60 (1):135-144.
    Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an o-minimal trace.
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  22.  1
    Ring Structure Theorems and Arithmetic Comprehension.Huishan Wu - 2021 - Archive for Mathematical Logic 60 (1-2):145-160.
    Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi ^{0}_{1}$$\end{document} subsets and show that Schur’s Lemma is provable in RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm RCA_{0}$$\end{document}. A ring (...)
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