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  1.  2
    Metrically Homogeneous Graphs of Diameter 3.Daniela A. Amato, Gregory Cherlin & H. Dugald Macpherson - 2020 - Journal of Mathematical Logic 21 (1):2050020.
    We classify countable metrically homogeneous graphs of diameter 3.
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  2.  6
    Tameness, Powerful Images, and Large Cardinals.Will Boney & Michael Lieberman - 2020 - Journal of Mathematical Logic 21 (1):2050024.
    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. AMS145(3) (2016) (...)
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  3. Bounds on Scott Ranks of Some Polish Metric Spaces.William Chan - 2020 - Journal of Mathematical Logic 21 (1):2150001.
    If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: (...)
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  4.  3
    Specializing Trees and Answer to a Question of Williams.Mohammad Golshani & Saharon Shelah - 2020 - Journal of Mathematical Logic 21 (1):2050023.
    We show that if [Formula: see text] then any nontrivial [Formula: see text]-closed forcing notion of size [Formula: see text] is forcing equivalent to [Formula: see text] the Cohen forcing for adding a new Cohen subset of [Formula: see text] We also produce, relative to the existence of suitable large cardinals, a model of [Formula: see text] in which [Formula: see text] and all [Formula: see text]-closed forcing notion of size [Formula: see text] collapse [Formula: see text] and hence are (...)
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  5.  2
    Turing Degrees in Polish Spaces and Decomposability of Borel Functions.Vassilios Gregoriades, Takayuki Kihara & Keng Meng Ng - 2020 - Journal of Mathematical Logic 21 (1):2050021.
    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 (...)
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  6.  4
    Knaster and Friends II: The C-Sequence Number.Chris Lambie-Hanson & Assaf Rinot - 2020 - Journal of Mathematical Logic 21 (1):2150002.
    Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong (...)
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  7.  1
    A Descriptive Main Gap Theorem.Francesco Mangraviti & Luca Motto Ros - 2020 - Journal of Mathematical Logic 21 (1):2050025.
    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 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by (...)
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  8.  1
    Pseudofinite Difference Fields and Counting Dimensions.Tingxiang Zou - 2020 - Journal of Mathematical Logic 21 (1):2050022.
    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.
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  9.  7
    Every Zero-Dimensional Homogeneous Space is Strongly Homogeneous Under Determinacy.Raphaël Carroy, Andrea Medini & Sandra Müller - 2020 - Journal of Mathematical Logic 20 (3):2050015.
    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, (...)
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  10.  2
    Defining Integer-Valued Functions in Rings of Continuous Definable Functions Over a Topological Field.Luck Darnière & Marcus Tressl - 2020 - Journal of Mathematical Logic 20 (3):2050014.
    Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or (...)
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  11.  7
    Local Saturation and Square Everywhere.Monroe Eskew - 2020 - Journal of Mathematical Logic 20 (3):2050019.
    We show that it is consistent relative to a huge cardinal that for all infinite cardinals [Formula: see text], [Formula: see text] holds and there is a stationary [Formula: see text] such that [Formula: see text] is [Formula: see text]-saturated.
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  12.  1
    Extender-Based Forcings with Overlapping Extenders and Negations of the Shelah Weak Hypothesis.Moti Gitik - 2020 - Journal of Mathematical Logic 20 (3):2050013.
    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.
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  13.  12
    Galois Groups as Quotients of Polish Groups.Krzysztof Krupiński & Tomasz Rzepecki - 2020 - Journal of Mathematical Logic 20 (3):2050018.
    We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F_σ 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 ∅. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and (...)
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  14.  3
    Constructing Sequences One Step at a Time.Henry Towsner - 2020 - Journal of Mathematical Logic 20 (3):2050017.
    We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain (CAC) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma (WKL), including showing that CAC+WKL 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 CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.
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  15.  2
    Coding in the Automorphism Group of a Computably Categorical Structure.Dan Turetsky - 2020 - Journal of Mathematical Logic 20 (3):2050016.
    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.
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  16.  3
    Weakly Minimal Groups with a New Predicate.Gabriel Conant & Michael C. Laskowski - 2020 - Journal of Mathematical Logic 20 (2):2050011.
    Fix a weakly minimal (i.e. superstable U-rank 1) structure M. Let M∗ be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M. We show that all formulas in the expansion (M∗,A) are equivalent to bounded formulas, and so (M,A) is stable (or NIP) if and only if the M-induced structure AM on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with (...)
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  17.  4
    The Ramsey Theory of the Universal Homogeneous Triangle-Free Graph.Natasha Dobrinen - 2020 - Journal of Mathematical Logic 20 (2):2050012.
    The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work (...)
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  18.  10
    Mice with Finitely Many Woodin Cardinals From Optimal Determinacy Hypotheses.Sandra Müller, Ralf Schindler & W. Hugh Woodin - 2020 - Journal of Mathematical Logic 20 (Supp01):1950013.
    We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...)
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