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  • Bounds on Scott ranks of some polish metric spaces
    J. Math. Log. (IF 1.318) Pub Date : 2020-07-15
    William Chan

    If 𝒩 is a proper Polish metric space and ℳ is any countable dense submetric space of 𝒩, then the Scott rank of 𝒩 in the natural first-order language of metric spaces is countable and in fact at most ω1ℳ+1, where ω1ℳ is the Church–Kleene ordinal of ℳ (construed as a subset of ω) which is the least ordinal with no presentation on ω computable from ℳ. If 𝒩 is a rigid Polish metric space and ℳ is any

    更新日期:2020-08-09
  • Mice with finitely many Woodin cardinals from optimal determinacy hypotheses
    J. Math. Log. (IF 1.318) Pub Date : 2020-07-27
    Sandra Müller; Ralf Schindler; W. Hugh Woodin

    We prove the following result which is due to the third author. Let n≥1. If Πn1 determinacy and Πn+11 determinacy both hold true and there is no Σn+21-definable ω1-sequence of pairwise distinct reals, then Mn# exists and is ω1-iterable. The proof yields that Πn+11 determinacy implies that Mn#(x) exists and is ω1-iterable for all reals x. A consequence is the Determinacy Transfer Theorem for arbitrary

    更新日期:2020-07-27
  • A descriptive Main Gap Theorem
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-24
    Francesco Mangraviti; Luca Motto Ros

    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 T and the Borel rank of the isomorphism relation ≅Tκ on its models of size κ, for κ any cardinal satisfying κ<κ=κ>2ℵ0

    更新日期:2020-07-24
  • Metrically homogeneous graphs of diameter 3
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-19
    Daniela A. Amato; Gregory Cherlin; H. Dugald Macpherson

    We classify countable metrically homogeneous graphs of diameter 3.

    更新日期:2020-07-24
  • Tameness, powerful images, and large cardinals
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-19
    Will Boney; Michael Lieberman

    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

    更新日期:2020-07-24
  • Specializing trees and answer to a question of Williams
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-13
    Mohammad Golshani; Saharon Shelah

    We show that if cf(2ℵ0)=ℵ1, then any nontrivial ℵ1-closed forcing notion of size ≤2ℵ0 is forcing equivalent to Add(ℵ1,1), the Cohen forcing for adding a new Cohen subset of ω1. We also produce, relative to the existence of suitable large cardinals, a model of ZFC in which 2ℵ0=ℵ2 and all ℵ1-closed forcing notion of size ≤2ℵ0 collapse ℵ2, and hence are forcing equivalent to Add(ℵ1,1). These results answer

    更新日期:2020-07-24
  • Turing degrees in Polish spaces and decomposability of Borel functions
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-04
    Vassilios Gregoriades; Takayuki Kihara; Keng Meng Ng

    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

    更新日期:2020-07-24
  • Pseudofinite difference fields and counting dimensions
    J. Math. Log. (IF 1.318) Pub Date : 2020-06-04
    Tingxiang Zou

    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.

    更新日期:2020-07-24
  • Constructing sequences one step at a time
    J. Math. Log. (IF 1.318) Pub Date : 2020-03-25
    Henry Towsner

    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”

    更新日期:2020-03-25
  • Local saturation and square everywhere
    J. Math. Log. (IF 1.318) Pub Date : 2020-03-25
    Monroe Eskew

    We show that it is consistent relative to a huge cardinal that for all infinite cardinals κ, □κ holds and there is a stationary S⊆κ+ such that NSκ+↾S is κ++-saturated.

    更新日期:2020-03-25
  • Galois groups as quotients of Polish groups
    J. Math. Log. (IF 1.318) Pub Date : 2020-03-10
    Krzysztof Krupiński; Tomasz Rzepecki

    We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an Fσ 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

    更新日期:2020-03-10
  • Every zero-dimensional homogeneous space is strongly homogeneous under determinacy
    J. Math. Log. (IF 1.318) Pub Date : 2020-03-04
    Raphaël Carroy; Andrea Medini; Sandra Müller

    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

    更新日期:2020-03-04
  • Coding in the automorphism group of a computably categorical structure
    J. Math. Log. (IF 1.318) Pub Date : 2020-02-18
    Dan Turetsky

    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.

    更新日期:2020-02-18
  • Extender-based forcings with overlapping extenders and negations of the Shelah Weak Hypothesis
    J. Math. Log. (IF 1.318) Pub Date : 2020-01-30
    Moti Gitik

    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.

    更新日期:2020-01-30
  • Defining integer-valued functions in rings of continuous definable functions over a topological field
    J. Math. Log. (IF 1.318) Pub Date : 2020-01-30
    Luck Darnière; Marcus Tressl

    Let 𝒦 be an expansion of either an ordered field (K,≤), or a valued field (K,v). Given a definable set X⊆Km let 𝒞(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure 𝒦, in particular when 𝒦 is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers ℤ is interpretable in 𝒞(X). If

    更新日期:2020-01-30
  • The Ramsey theory of the universal homogeneous triangle-free graph
    J. Math. Log. (IF 1.318) Pub Date : 2020-01-28
    Natasha Dobrinen

    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.

    更新日期:2020-01-28
  • Weakly minimal groups with a new predicate
    J. Math. Log. (IF 1.318) Pub Date : 2020-01-22
    Gabriel Conant; Michael C. Laskowski

    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

    更新日期:2020-01-22
  • Model theory of Steiner triple systems
    J. Math. Log. (IF 1.318) Pub Date : 2019-12-31
    Silvia Barbina; Enrique Casanovas

    A Steiner triple system (STS) is a set S together with a collection ℬ of subsets of S of size 3 such that any two elements of S belong to exactly one element of ℬ. It is well known that the class of finite STS has a Fraïssé limit MF. Here, we show that the theory TSq∗ of MF is the model completion of the theory of STSs. We also prove that TSq∗ is not small and it has quantifier elimination, TP2, NSOP1

    更新日期:2019-12-31
  • Locally definable subgroups of semialgebraic groups
    J. Math. Log. (IF 1.318) Pub Date : 2019-12-16
    Elías Baro; Pantelis E. Eleftheriou; Ya’acov Peterzil

    We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let G be an abelian semialgebraic group over a real closed field R and let X be a semialgebraic subset of G. Then the group generated by X contains a generic set and, if connected, it is divisible. More generally,

    更新日期:2019-12-16
  • Logical laws for short existential monadic second-order sentences about graphs
    J. Math. Log. (IF 1.318) Pub Date : 2019-12-12
    M. E. Zhukovskii

    In 2001, Le Bars proved that there exists an existential monadic second-order (EMSO) sentence such that the probability that it is true on G(n,p=1/2) does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for p∈{3−52,5−12}, and give new examples of sentences with fewer variables without convergence

    更新日期:2019-12-12
  • Exponential-constructible functions in P-minimal structures
    J. Math. Log. (IF 1.318) Pub Date : 2019-11-27
    Saskia Chambille; Pablo Cubides Kovacsics; Eva Leenknegt

    Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of P-minimal

    更新日期:2019-11-27
  • Definable V-topologies, Henselianity and NIP
    J. Math. Log. (IF 1.318) Pub Date : 2019-11-15
    Yatir Halevi; Assaf Hasson; Franziska Jahnke

    We initiate the study of definable V-topologies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K,v1,v2) is a bi-valued NIP field with v1 henselian (respectively, t-henselian), then v1 and v2 are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields.

    更新日期:2019-11-15
  • Computable aspects of the Bachmann–Howard principle
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-08
    Anton Freund

    We have previously established that Π11-comprehension is equivalent to the statement that every dilator has a well-founded Bachmann–Howard fixed point, over ATR0. In this paper, we show that the base theory can be lowered to RCA0. We also show that the minimal Bachmann–Howard fixed point of a dilator T can be represented by a notation system 𝜗(T), which is computable relative to T. The statement that

    更新日期:2019-10-08
  • Computing from projections of random points
    J. Math. Log. (IF 1.318) Pub Date : 2019-09-13
    Noam Greenberg; Joseph S. Miller; André Nies

    We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call 1/2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function

    更新日期:2019-09-13
  • Uncountable structures are not classifiable up to bi-embeddability
    J. Math. Log. (IF 1.318) Pub Date : 2019-09-06
    Filippo Calderoni; Heike Mildenberger; Luca Motto Ros

    Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic164(12) (2013) 1454–1492], we show that whenever κ is a cardinal satisfying κ<κ=κ>ω, then the embeddability relation between κ-sized structures is strongly invariantly universal, and hence complete for (κ-)analytic quasi-orders

    更新日期:2019-09-06
  • The consistency strength of hyperstationarity
    J. Math. Log. (IF 1.318) Pub Date : 2019-09-06
    Joan Bagaria; Menachem Magidor; Salvador Mancilla

    We introduce the large-cardinal notions of ξ-greatly-Mahlo and ξ-reflection cardinals and prove (1) in the constructible universe, L, the first ξ-reflection cardinal, for ξ a successor ordinal, is strictly between the first ξ-greatly-Mahlo and the first Πξ1-indescribable cardinals, (2) assuming the existence of a ξ-reflection cardinal κ in L, ξ a successor ordinal, there exists a forcing notion in

    更新日期:2019-09-06
  • The special Aronszajn tree property
    J. Math. Log. (IF 1.318) Pub Date : 2019-08-23
    Mohammad Golshani; Yair Hayut

    Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal κ, there are κ+-Aronszajn trees, and all such trees are special.

    更新日期:2019-08-23
  • Recursive functions and existentially closed structures
    J. Math. Log. (IF 1.318) Pub Date : 2019-08-08
    Emil Jeřábek

    The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson’s theory R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion

    更新日期:2019-08-08
  • Rado’s Conjecture and its Baire version
    J. Math. Log. (IF 1.318) Pub Date : 2019-07-24
    Jing Zhang

    Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height ω1 has a nonspecial subtree of size ℵ1. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular

    更新日期:2019-07-24
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