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Authors:Sandra Müller, Ralf Schindler, W. Hugh Woodin Abstract: Journal of Mathematical Logic, Volume 20, Issue Supp01, October 2020. We prove the following result which is due to the third author. Let [math]. If [math] determinacy and [math] determinacy both hold true and there is no [math]-definable [math]-sequence of pairwise distinct reals, then [math] exists and is [math]-iterable. The proof yields that [math] determinacy implies that [math] exists and is [math]-iterable for all reals [math]. A consequence is the Determinacy Transfer Theorem for arbitrary [math], namely the statement that [math] determinacy implies [math] determinacy. Citation: Journal of Mathematical Logic PubDate: 2020-07-27T07:00:00Z DOI: 10.1142/S0219061319500132 Issue No:Vol. 20, No. Supp01 (2020)

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Authors:Paul McKenney, Alessandro Vignati Abstract: Journal of Mathematical Logic, Ahead of Print. We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [math]-algebras with the metric approximation property and an increasing approximate identity of projections. Citation: Journal of Mathematical Logic PubDate: 2020-10-23T07:00:00Z DOI: 10.1142/S0219061321500069

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Authors:Rahman Mohammadpour, Boban Veličković Abstract: Journal of Mathematical Logic, Ahead of Print. Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [math] holds. This principle implies [math] and [math], and hence the tree property at [math] and [math], the Singular Cardinal Hypothesis, and the failure of the weak square principle [math], for all regular [math]. In addition, it implies that the restriction of the approachability ideal [math] to the set of ordinals of cofinality [math] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell. Citation: Journal of Mathematical Logic PubDate: 2020-10-06T07:00:00Z DOI: 10.1142/S0219061321500033

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Authors:Artem Chernikov, Sergei Starchenko Abstract: Journal of Mathematical Logic, Ahead of Print. We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups' (and how to use them in Erdos geometry'), Combinatorica 32(5) 537–571 (2012)] for relations definable in strongly minimal structures that are interpretable in distal structures. Citation: Journal of Mathematical Logic PubDate: 2020-10-06T07:00:00Z DOI: 10.1142/S0219061321500045

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Authors:Antongiulio Fornasiero, Elliot Kaplan Abstract: Journal of Mathematical Logic, Ahead of Print. Let [math] be a complete, model complete o-minimal theory extending the theory [math] of real closed ordered fields in some appropriate language [math]. We study derivations [math] on models [math]. We introduce the notion of a [math]-derivation: a derivation which is compatible with the [math]-definable [math]-functions on [math]. We show that the theory of [math]-models with a [math]-derivation has a model completion [math]. The derivation in models [math] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [math]-substructure of [math]. If [math], then [math] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [math] has [math] as its open core, that [math] is distal, and that [math] eliminates imaginaries. We also show that the theory of [math]-models with finitely many commuting [math]-derivations has a model completion. Citation: Journal of Mathematical Logic PubDate: 2020-10-06T07:00:00Z DOI: 10.1142/S0219061321500070

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Authors:Chris Lambie-Hanson, Assaf Rinot Abstract: Journal of Mathematical Logic, Ahead of Print. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [math]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [math] and independence results about the [math]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [math]-sequence spectrum and uncover some tight connections between the [math]-sequence spectrum and the strong coloring principle [math], introduced in Part I of this series. Citation: Journal of Mathematical Logic PubDate: 2020-08-19T07:00:00Z DOI: 10.1142/S0219061321500021

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Authors:William Chan Abstract: Journal of Mathematical Logic, Ahead of Print. If [math] is a proper Polish metric space and [math] is any countable dense submetric space of [math], then the Scott rank of [math] in the natural first-order language of metric spaces is countable and in fact at most [math], where [math] is the Church–Kleene ordinal of [math] (construed as a subset of [math]) which is the least ordinal with no presentation on [math] computable from [math]. If [math] is a rigid Polish metric space and [math] is any countable dense submetric space, then the Scott rank of [math] is countable and in fact less than [math]. Citation: Journal of Mathematical Logic PubDate: 2020-07-15T07:00:00Z DOI: 10.1142/S021906132150001X

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Authors:Francesco Mangraviti, Luca Motto Ros Abstract: Journal of Mathematical Logic, Ahead of Print. Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [math] and the Borel rank of the isomorphism relation [math] on its models of size [math], for [math] any cardinal satisfying [math]. This is achieved by establishing a link between said rank and the [math]-Scott height of the [math]-sized models of [math], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [math], either [math] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [math]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [math], and provide a characterization of categoricity of [math] in terms of the descriptive set-theoretical complexity of [math]. Citation: Journal of Mathematical Logic PubDate: 2020-06-24T07:00:00Z DOI: 10.1142/S0219061320500257

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Authors:Daniela A. Amato, Gregory Cherlin, H. Dugald Macpherson Abstract: Journal of Mathematical Logic, Ahead of Print. We classify countable metrically homogeneous graphs of diameter 3. Citation: Journal of Mathematical Logic PubDate: 2020-06-19T07:00:00Z DOI: 10.1142/S0219061320500208

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Authors:Will Boney, Michael Lieberman Abstract: Journal of Mathematical Logic, Ahead of Print. We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648]. Citation: Journal of Mathematical Logic PubDate: 2020-06-19T07:00:00Z DOI: 10.1142/S0219061320500245

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Authors:Mohammad Golshani, Saharon Shelah Abstract: Journal of Mathematical Logic, Ahead of Print. We show that if [math] then any nontrivial [math]-closed forcing notion of size [math] is forcing equivalent to [math] the Cohen forcing for adding a new Cohen subset of [math] We also produce, relative to the existence of suitable large cardinals, a model of ZFC in which [math] and all [math]-closed forcing notion of size [math] collapse [math] and hence are forcing equivalent to [math] These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman–Magidor–Shelah by showing that it is consistent that every partial order which adds a new subset of [math] collapses [math] or [math] Citation: Journal of Mathematical Logic PubDate: 2020-06-13T07:00:00Z DOI: 10.1142/S0219061320500233

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Authors:Vassilios Gregoriades, Takayuki Kihara, Keng Meng Ng Abstract: Journal of Mathematical Logic, Ahead of Print. We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. Citation: Journal of Mathematical Logic PubDate: 2020-06-04T07:00:00Z DOI: 10.1142/S021906132050021X

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Authors:Tingxiang Zou Abstract: Journal of Mathematical Logic, Ahead of Print. We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields. Citation: Journal of Mathematical Logic PubDate: 2020-06-04T07:00:00Z DOI: 10.1142/S0219061320500221

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Authors:Henry Towsner Abstract: Journal of Mathematical Logic, Ahead of Print. We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain ([math]) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma ([math]), including showing that [math] does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between [math] and the Ascending/Descending Sequences principle, even in the presence of [math]. Citation: Journal of Mathematical Logic PubDate: 2020-03-25T07:00:00Z DOI: 10.1142/S0219061320500178

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Authors:Monroe Eskew Abstract: Journal of Mathematical Logic, Ahead of Print. We show that it is consistent relative to a huge cardinal that for all infinite cardinals [math], [math] holds and there is a stationary [math] such that [math] is [math]-saturated. Citation: Journal of Mathematical Logic PubDate: 2020-03-25T07:00:00Z DOI: 10.1142/S0219061320500191

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Authors:Krzysztof Krupiński, Tomasz Rzepecki Abstract: Journal of Mathematical Logic, Ahead of Print. We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an [math] normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over [math]. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932] which says that for any strong type defined on a single complete type over [math], smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups. Citation: Journal of Mathematical Logic PubDate: 2020-03-10T07:00:00Z DOI: 10.1142/S021906132050018X

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Authors:Raphaël Carroy, Andrea Medini, Sandra Müller Abstract: Journal of Mathematical Logic, Ahead of Print. All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev. Citation: Journal of Mathematical Logic PubDate: 2020-03-04T08:00:00Z DOI: 10.1142/S0219061320500154

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Authors:Dan Turetsky Abstract: Journal of Mathematical Logic, Ahead of Print. Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank. Citation: Journal of Mathematical Logic PubDate: 2020-02-18T08:00:00Z DOI: 10.1142/S0219061320500166

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Authors:Moti Gitik Abstract: Journal of Mathematical Logic, Ahead of Print. Extender-based Prikry–Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed. Citation: Journal of Mathematical Logic PubDate: 2020-01-30T08:00:00Z DOI: 10.1142/S0219061320500130

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Authors:Luck Darnière, Marcus Tressl Abstract: Journal of Mathematical Logic, Ahead of Print. Let [math] be an expansion of either an ordered field [math], or a valued field [math]. Given a definable set [math] let [math] be the ring of continuous definable functions from [math] to [math]. Under very mild assumptions on the geometry of [math] and on the structure [math], in particular when [math] is [math]-minimal or [math]-minimal, or an expansion of a local field, we prove that the ring of integers [math] is interpretable in [math]. If [math] is [math]-minimal and [math] is definably connected of pure dimension [math], then [math] defines the subring [math]. If [math] is [math]-minimal and [math] has no isolated points, then there is a discrete ring [math] contained in [math] and naturally isomorphic to [math], such that the ring of functions [math] which take values in [math] is definable in [math]. Citation: Journal of Mathematical Logic PubDate: 2020-01-30T08:00:00Z DOI: 10.1142/S0219061320500142