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  1.  6
    Maximal Trees.Jörg Brendle - 2018 - Archive for Mathematical Logic 57 (3-4):421-428.
    We show that, consistently, there can be maximal subtrees of \\) and \ / {\mathrm {fin}}\) of arbitrary regular uncountable size below the size of the continuum \. We also show that there are no maximal subtrees of \ / {\mathrm {fin}}\) with countable levels. Our results answer several questions of Campero-Arena et al..
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  2.  2
    Some Remarks on Baire’s Grand Theorem.Riccardo Camerlo & Jacques Duparc - 2018 - Archive for Mathematical Logic 57 (3-4):195-201.
    We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to \ that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.
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  3.  1
    A Weak Variant of Hindman’s Theorem Stronger Than Hilbert’s Theorem.Lorenzo Carlucci - 2018 - Archive for Mathematical Logic 57 (3-4):381-389.
    Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound (...)
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  4.  1
    Implicational Logics III: Completeness Properties.Petr Cintula & Carles Noguera - 2018 - Archive for Mathematical Logic 57 (3-4):391-420.
    This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in (...)
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  5.  1
    The Subcompleteness of Magidor Forcing.Gunter Fuchs - 2018 - Archive for Mathematical Logic 57 (3-4):273-284.
    It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length \ sequence of normal ultrafilters, increasing in the Mitchell order, to \, is subcomplete.
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  6.  1
    Interpretable Groups in Mann Pairs.Haydar Göral - 2018 - Archive for Mathematical Logic 57 (3-4):203-237.
    In this paper, we study an algebraically closed field \ expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \\). This enables us to characterize the interpretable groups when \ is divisible. Every interpretable group H in \\) is, up to (...)
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  7.  1
    Countable OD Sets of Reals Belong to the Ground Model.Vladimir Kanovei & Vassily Lyubetsky - 2018 - Archive for Mathematical Logic 57 (3-4):285-298.
    It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.
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  8.  2
    Scott Sentences for Certain Groups.Julia F. Knight & Vikram Saraph - 2018 - Archive for Mathematical Logic 57 (3-4):453-472.
    We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable \ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d-\” sentence and a (...)
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  9.  3
    Decidability of the AE-Theory of the Lattice of $${\varPi }_1^0$$ Π 1 0 Classes.Linda Lawton - 2018 - Archive for Mathematical Logic 57 (3-4):429-451.
    An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \, \cap, \cup, 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.
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  10.  2
    Expressivity in Chain-Based Modal Logics.Michel Marti & George Metcalfe - 2018 - Archive for Mathematical Logic 57 (3-4):361-380.
    We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.
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  11.  2
    On Katětov and Katětov–Blass Orders on Analytic P-Ideals and Borel Ideals.Hiroshi Sakai - 2018 - Archive for Mathematical Logic 57 (3-4):317-327.
    Minami–Sakai :883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders. In the course of the proof of (...)
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  12.  1
    Continuous Reducibility and Dimension of Metric Spaces.Philipp Schlicht - 2018 - Archive for Mathematical Logic 57 (3-4):329-359.
    If is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space of positive dimension, there are uncountably many Borel subsets of that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \\) is called the Wadge quasi-order (...)
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  13.  1
    L -Groups C in Continuous Logic.Philip Scowcroft - 2018 - Archive for Mathematical Logic 57 (3-4):239-272.
    In the context of continuous logic, this paper axiomatizes both the class \ of lattice-ordered groups isomorphic to C for X compact and the subclass \ of structures existentially closed in \; shows that the theory of \ is \-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \ and \; shows that \\in \mathcal {C}\) has a prime-model extension in \ just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas (...)
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  14.  1
    Quasiminimal Abstract Elementary Classes.Sebastien Vasey - 2018 - Archive for Mathematical Logic 57 (3-4):299-315.
    We propose the notion of a quasiminimal abstract elementary class. This is an AEC satisfying four semantic conditions: countable Löwenheim–Skolem–Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber’s quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC, and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular (...)
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  15.  4
    Aronszajn and Kurepa Trees.James Cummings - 2018 - Archive for Mathematical Logic 57 (1-2):83-90.
    Monroe Eskew and \, 2016. https://mathoverflow.net/questions/217951/tree-properties-on-omega-1-and-omega-2) asked whether the tree property at \ implies there is no Kurepa tree. We prove that the tree property at \ is consistent with the existence of \-trees with as many branches as desired.
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  16.  2
    Largest Initial Segments Pointwise Fixed by Automorphisms of Models of Set Theory.Ali Enayat, Matt Kaufmann & Zachiri McKenzie - 2018 - Archive for Mathematical Logic 57 (1-2):91-139.
    Given a model \ of set theory, and a nontrivial automorphism j of \, let \\) be the submodel of \ whose universe consists of elements m of \ such that \=x\) for every x in the transitive closure of m ). Here we study the class \ of structures of the form \\), where the ambient model \ satisfies a frugal yet robust fragment of \ known as \, and \=m\) whenever m is a finite ordinal in the sense (...)
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  17.  3
    Preface.Ali Enayat, Massoud Pourmahdian & Ralf Schindler - 2018 - Archive for Mathematical Logic 57 (1-2):1-2.
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  18.  2
    The Small Index Property for Homogeneous Models in AEC's.Zaniar Ghadernezhad & Andrés Villaveces - 2018 - Archive for Mathematical Logic 57 (1-2):141-157.
    We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah. We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.
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  19.  2
    The Tree Property at the Successor of a Singular Limit of Measurable Cardinals.Mohammad Golshani - 2018 - Archive for Mathematical Logic 57 (1-2):3-25.
    Assume \ is a singular limit of \ supercompact cardinals, where \ is a limit ordinal. We present two methods for arranging the tree property to hold at \ while making \ the successor of the limit of the first \ measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \ with the failure of SCH at \. This extends results of Neeman and Sinapova. The second method is also used to (...)
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  20.  2
    On a Question of Silver About Gap-Two Cardinal Transfer Principles.Mohammad Golshani & Shahram Mohsenipour - 2018 - Archive for Mathematical Logic 57 (1-2):27-35.
    Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel’s constructible universe L, in which the \ holds and the transfer principles \ \rightarrow \) and \ \rightarrow \) fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
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  21.  3
    Model Theory of Finite and Pseudofinite Groups.Dugald Macpherson - 2018 - Archive for Mathematical Logic 57 (1-2):159-184.
    This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.
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  22.  4
    On Some Dynamical Aspects of NIP Theories.Alireza Mofidi - 2018 - Archive for Mathematical Logic 57 (1-2):37-71.
    We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems (...)
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  23.  3
    The Long Extender Algebra.Ralf Schindler - 2018 - Archive for Mathematical Logic 57 (1-2):73-82.
    Generalizing Woodin’s extender algebra, cf. e.g. Steel Handbook of set theory, Springer, Berlin, 2010), we isolate the long extender algebra as a general version of Bukowský’s forcing, cf. Bukovský, in the presence of a supercompact cardinal.
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  24.  5
    Collapsing $$Omega _2$$ with Semi-Proper Forcing.Stevo Todorcevic - 2018 - Archive for Mathematical Logic 57 (1-2):185-194.
    We examine the differences between three standard classes of forcing notions relative to the way they collapse the continuum. It turns out that proper and semi-proper posets behave differently in that respect from the class of posets that preserve stationary subsets of \.
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