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  • Generic derivations on o-minimal structures
    J. Math. Log. (IF 1.318) Pub Date : 2020-10-06
    Antongiulio Fornasiero; Elliot Kaplan

    Let T be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language L. We study derivations δ on models ℳ⊧T. We introduce the notion of a T-derivation: a derivation which is compatible with the L(∅)-definable 𝒞1-functions on ℳ. We show that the theory of T-models with a T-derivation has a model completion TGδ. The derivation in models

    更新日期:2020-10-21
  • Guessing models and the approachability ideal
    J. Math. Log. (IF 1.318) Pub Date : 2020-10-06
    Rahman Mohammadpour; Boban Veličković

    Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call GM+(ω3,ω1) holds. This principle implies ISP(ω2) and ISP(ω3), and hence the tree property at ω2 and ω3, the Singular Cardinal Hypothesis, and the failure of the weak square principle □(ω2,λ), for all regular λ≥ω2. In addition, it implies that the restriction of the approachability

    更新日期:2020-10-07
  • Model-theoretic Elekes–Szabó in the strongly minimal case
    J. Math. Log. (IF 1.318) Pub Date : 2020-10-06
    Artem Chernikov; Sergei Starchenko

    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?), Combinatorica32(5) 537–571 (2012)] for relations definable in strongly minimal structures that are interpretable in distal structures.

    更新日期:2020-10-07
  • 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:2020-09-30
  • Knaster and friends II: The C-sequence number
    J. Math. Log. (IF 1.318) Pub Date : 2020-08-19
    Chris Lambie-Hanson; Assaf Rinot

    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

    更新日期:2020-08-24
  • 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
  • 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
  • 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:2020-07-15
  • 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:2020-06-19
  • 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:2020-06-04
  • 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:2020-06-04
  • 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
  • Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Libor Barto; Michael Kompatscher; Miroslav Olšák; Van Pham Trung; Michael Pinsker

    There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain nontrivial linear identity modulo outer embeddings. The second conjecture, challenging the approach

    更新日期:2019-10-02
  • Hypergraphs and proper forcing
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Jindřich Zapletal

    Given a Polish space X and a countable collection of analytic hypergraphs on X, I consider the σ-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

    更新日期:2019-10-02
  • Ordinal definability and combinatorics of equivalence relations
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    William Chan

    Assume ZF+AD++V=L(𝒫(ℝ)). Let E be a Σ11 equivalence relation coded in HOD. E has an ordinal definable equivalence class without any ordinal definable elements if and only if HOD⊧E is unpinned. ZF+AD++V=L(𝒫(ℝ)) proves E-class section uniformization when E is a Σ11 equivalence relation on ℝ which is pinned in every transitive model of ZFC containing the real which codes E: Suppose R is a relation on

    更新日期:2019-10-02
  • Pseudofinite difference fields
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Tingxiang Zou

    The author requires to retract this paper because there is a gap in the proof of Lemma 3.2, hence also in Theorem 3.1. The revised version is in preparation.

    更新日期:2019-10-02
  • Boundedness and absoluteness of some dynamical invariants in model theory
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Krzysztof Krupiński; Ludomir Newelski; Pierre Simon

    Let ℭ be a monster model of an arbitrary theory T, let ᾱ be any (possibly infinite) tuple of bounded length of elements of ℭ, and let c̄ be an enumeration of all elements of ℭ (so a tuple of unbounded length). By Sᾱ(ℭ) we denote the compact space of all complete types over ℭ extending tp(ᾱ/∅), and Sc̄(ℭ) is defined analogously. Then Sᾱ(ℭ) and Sc̄(ℭ) are naturally Aut(ℭ)-flows (even Aut(ℭ)-ambits)

    更新日期:2019-10-02
  • On possible restrictions of the null ideal
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Ashutosh Kumar; Saharon Shelah

    We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size ℵ1, such that every transversal of this partition

    更新日期:2019-10-02
  • Amalgamation of types in pseudo-algebraically closed fields and applications
    J. Math. Log. (IF 1.318) Pub Date : 2019-10-02
    Zoé Chatzidakis

    This paper studies unbounded pseudo-algebraically closed (PAC) fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.

    更新日期:2019-10-02
  • 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
  • On the mathematical and foundational significance of the uncountable
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Dag Normann; Sam Sanders

    We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable

    更新日期:2019-06-11
  • Independence in randomizations
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Uri Andrews; Isaac Goldbring; H. Jerome Keisler

    The randomization of a complete first-order theory T is the complete continuous theory TR with two sorts, a sort for random elements of models of T and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of T. We show that if T has the exchange property and acl=dcl, then TR has a strict independence relation

    更新日期:2019-06-11
  • Linear orders: When embeddability and epimorphism agree
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Riccardo Camerlo; Raphaël Carroy; Alberto Marcone

    When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is a Ď2(Π11)-complete set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders.

    更新日期:2019-06-11
  • Diamonds, compactness, and measure sequences
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Omer Ben-Neria

    We establish the consistency of the failure of the diamond principle on a cardinal κ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of κ in a Radin generic extension.

    更新日期:2019-06-11
  • A model of second-order arithmetic satisfying AC but not DC
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Sy-David Friedman; Victoria Gitman; Vladimir Kanovei

    We show that there is a β-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a Π21-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ZFC−. This work is a rediscovery by the first two authors of a result

    更新日期:2019-06-11
  • Connected choice and the Brouwer fixed point theorem
    J. Math. Log. (IF 1.318) Pub Date : 2019-06-11
    Vasco Brattka; Stéphane Le Roux; Joseph S. Miller; Arno Pauly

    We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the

    更新日期:2019-06-11
  • The Bristol model: An abyss called a Cohen real
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Asaf Karagila

    We construct a model M of ZF which lies between L and L[c] for a Cohen real c and does not have the form L(x) for any set x. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of ZF,

    更新日期:2018-11-20
  • Omitting types in logic of metric structures
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Ilijas Farah; Menachem Magidor

    This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model

    更新日期:2018-11-20
  • There is no classification of the decidably presentable structures
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Matthew Harrison-Trainor

    A computable structure 𝒜 is decidable if, given a formula φ(x̄) of elementary first-order logic, and a tuple ā∈𝒜, we have a decision procedure to decide whether φ holds of ā. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is Σ11-complete. We also show that for

    更新日期:2018-11-20
  • The canonical topology on dp-minimal fields
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Will Johnson

    We construct a nontrivial definable type V field topology on any dp-minimal field K that is not strongly minimal, and prove that definable subsets of Kn have small boundary. Using this topology and its properties, we show that in any dp-minimal field K, dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and |K×/(K×)n| is

    更新日期:2018-11-20
  • No Tukey reduction of Lebesgue null to Silver null sets
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Otmar Spinas

    We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math.211 (2016) 473–480] is Tukey reducible to the Silver ideal.

    更新日期:2018-11-20
  • Toward a stability theory of tame abstract elementary classes
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Sebastien Vasey

    We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior

    更新日期:2018-11-20
  • Randomized feasible interpolation and monotone circuits with a local oracle
    J. Math. Log. (IF 1.318) Pub Date : 2018-11-20
    Jan Krajíček

    The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two NP sets U and V a small communication protocol (a general dag-like protocol

    更新日期:2018-11-20
  • Degrees containing members of thin Π10 classes are dense and co-dense
    J. Math. Log. (IF 1.318) Pub Date : 2018-06-03
    Rodney G. Downey; Guohua Wu; Yue Yang

    In [Countable thin Π10 classes, Ann. Pure Appl. Logic59 (1993) 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees (not necessarily c.e.) containing members of countable thin Π10 classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin Π10 classes. We will prove in this paper that the c.e. degrees containing no members of thin

    更新日期:2018-06-03
  • The linearity of the Mitchell order
    J. Math. Log. (IF 1.318) Pub Date : 2018-06-03
    Gabriel Goldberg

    We show from an abstract comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on sufficiently strong ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. This gives a conditional answer to the well-known question of whether a 2κ-supercompact cardinal κ must carry more than one normal measure of order 0. Conditioned on

    更新日期:2018-06-03
  • Torsion-free abelian groups with optimal Scott families
    J. Math. Log. (IF 1.318) Pub Date : 2018-06-03
    Alexander G. Melnikov

    We prove that for any computable successor ordinal of the formα=δ+2k(δlimit andk∈ω)there exists computable torsion-free abelian group(TFAG)that is relativelyΔα0-categorical and notΔα−10-categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σαc infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly Σαc-Scott

    更新日期:2018-06-03
  • A long chain of P-points
    J. Math. Log. (IF 1.318) Pub Date : 2018-06-03
    Borisa Kuzeljevic; Dilip Raghavan

    The notion of a δ-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each δ<ω2, any δ-generic sequence of P-points can be extended to an ω2-generic sequence. This shows that the CH implies that there is a chain of P-points of length 𝔠+ with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question

    更新日期:2018-06-03
  • Existentially closed fields with finite group actions
    J. Math. Log. (IF 1.318) Pub Date : 2018-06-03
    Daniel M. Hoffmann; Piotr Kowalski

    We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.

    更新日期:2018-06-03
  • Simultaneous stationary reflection and square sequences
    J. Math. Log. (IF 1.318) Pub Date : 2017-11-27
    Yair Hayut; Chris Lambie-Hanson

    We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

    更新日期:2017-11-27
  • Fraïssé’s conjecture in Π11-comprehension
    J. Math. Log. (IF 1.318) Pub Date : 2017-11-27
    Antonio Montalbán

    We prove Fraïssé’s conjecture within the system of Π11-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the Δ20-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the Δ20-bqo-ness of 3 is a Π21-statement strictly weaker than Π11-comprehension.

    更新日期:2017-11-27
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