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  • Orders on computable rings
    Math. Logic Q. (IF 0.244) Pub Date : 2020-07-12
    Huishan Wu

    The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to WKL 0 over RCA 0 . We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to WKL 0 over RCA 0 . In the theory of orderings on rings, following an idea of Serre, we often show the existence

    更新日期:2020-07-17
  • Stability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaces
    Math. Logic Q. (IF 0.244) Pub Date : 2020-07-01
    Karim Khanaki

    We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, Talagrand's stability , and explain the relationship between this property and the NIP in continuous logic. Using a result of Bourgain, Fremlin

    更新日期:2020-07-17
  • Singly generated quasivarieties and residuated structures
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-21
    Tommaso Moraschini; James G. Raftery; Johann J. Wannenburg

    A quasivariety K of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A . It is structurally complete if and only if the free ℵ0‐generated algebra in K can serve as A . A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if

    更新日期:2020-07-17
  • More on trees and Cohen reals
    Math. Logic Q. (IF 0.244) Pub Date : 2020-07-02
    Giorgio Laguzzi; Brendan Stuber‐Rousselle

    In this paper we analyse some questions concerning trees on κ, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in [6, Question 5.2] about the diagram for regularity properties.

    更新日期:2020-07-17
  • The classification of countable models of set theory
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-29
    John Clemens; Samuel Coskey; Samuel Dworetzky

    We study the complexity of the classification problem for countable models of set theory ( ZFC ). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of ZFC .

    更新日期:2020-07-17
  • A game‐theoretic proof of Shelah's theorem on labeled trees
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-16
    Trevor M. Wilson

    We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality κ of the family is much larger (in the sense of large cardinals) than the cardinality λ of the set of labels, more precisely if the partition relation κ → ( ω ) λ < ω holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of

    更新日期:2020-07-17
  • Axiomatizing first order consequences in inclusion logic
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-23
    Fan Yang

    Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed‐point logic. Inclusion logic is not axiomatisable in full, but its first order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first

    更新日期:2020-07-17
  • The Cichoń diagram for degrees of relative constructibility
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-23
    Corey Bacal Switzer

    Following a line of research initiated in [4], we describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cichoń diagram for cardinal characteristics of the continuum. We show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application, we

    更新日期:2020-07-17
  • On a strengthening of Jónssonness for ℵω
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-14
    Monroe Eskew

    We discuss a system of strengthenings of “ ℵ ω is Jónsson” indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.

    更新日期:2020-07-17
  • The choice property in tame expansions of o‐minimal structures
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-23
    Pantelis E. Eleftheriou; Ayhan Günaydın; Philipp Hieronymi

    We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o‐minimal structures. Most noteworthily, this property holds for dense pairs of real closed fields, as well as for expansions of o‐minimal structures by a dense independent set.

    更新日期:2020-07-17
  • Turing invariant sets and the perfect set property
    Math. Logic Q. (IF 0.244) Pub Date : 2020-06-14
    Clovis Hamel; Haim Horowitz; Saharon Shelah

    We show that ZF + DC + “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

    更新日期:2020-07-17
  • Editorial
    Math. Logic Q. (IF 0.244) Pub Date : 2020-04-02

    Dear readers of Mathematical Logic Quarterly: Dr Hugo Nobrega has been the Editorial Assistant of our journal for four years, from 2016 to 2019, and has now handed over to his successor, Dr Thomas Piecha from the Eberhard Karls Universität Tübingen, who started to work for the journal on New Year's Day 2020. We thank Hugo for four years of excellent service for the journal and welcome Thomas as our

    更新日期:2020-04-02
  • The torsion‐free part of the Ziegler spectrum of orders over Dedekind domains
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-24
    Lorna Gregory; Sonia L'Innocente; Carlo Toffalori

    We study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.

    更新日期:2020-03-24
  • Computability of graphs
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-19
    Zvonko Iljazović

    We consider topological pairs ( A , B ) , B ⊆ A , which have computable type, which means that they have the following property: if X is a computable topological space and f : A → X a topological imbedding such that f ( A ) and f ( B ) are semicomputable sets in X, then f ( A ) is a computable set in X. It is known, e.g., that ( M , ∂ M ) has computable type if M is a compact manifold with boundary

    更新日期:2020-03-19
  • On the Herbrand functional interpretation
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-16
    Paulo Oliva; Chuangjie Xu

    We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation

    更新日期:2020-03-16
  • Local weak presaturation of the strongly non‐stationary ideal
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-16
    Masahiro Shioya; Naoki Yamaura

    We give a model of set theory in which the strongly non‐stationary ideal over ℘ μ μ is weakly presaturated below some canonical set. Here μ is a regular uncountable cardinal. The model is the forcing extension with the Lévy collapse of a Woodin cardinal to the successor of μ. This improves on results of Goldring and of the first author.

    更新日期:2020-03-16
  • Lowness for isomorphism, countable ideals, and computable traceability
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-16
    Johanna N. Y. Franklin; Reed Solomon

    We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune free degrees, lowness for isomorphism is entirely independent of computable traceability.

    更新日期:2020-03-16
  • On weak square, approachability, the tree property, and failures of SCH in a choiceless context
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-16
    Arthur W. Apter

    We show that the consistency of the theories ZF + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “both □ ℵ ω ∗ and AP ℵ ω fail” and ZF + ¬ AC ω + “ GCH holds below ℵ ω ” + “there is an injection f : ℵ ω + 2 → ℘ ( ℵ ω ) ” + “ ℵ ω + 1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context

    更新日期:2020-03-16
  • Expansions of the p‐adic numbers that interpret the ring of integers
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-11
    Nathanaël Mariaule

    Let Q p ∼ be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of Q p ∼ expanded by two predicates interpreted by multiplicative subgroups α Z and β Z where α , β ∈ N are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that

    更新日期:2020-03-11
  • Cohen forcing and inner models
    Math. Logic Q. (IF 0.244) Pub Date : 2020-03-09
    Jonas Reitz

    Given an inner model W ⊂ V and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model Add ( κ , 1 ) W . The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality

    更新日期:2020-03-09
  • On p‐adic semi‐algebraic continuous selections
    Math. Logic Q. (IF 0.244) Pub Date : 2020-02-25
    Athipat Thamrongthanyalak

    Let E ⊆ Q p n and T be a set‐valued map from E to Q p m . We prove that if T is p‐adic semi‐algebraic, lower semi‐continuous and T ( x ) is closed for every x ∈ E , then T has a p‐adic semi‐algebraic continuous selection. In addition, we include three applications of this result. The first one is related to Fefferman's and Kollár's question on existence of p‐adic semi‐algebraic continuous solution

    更新日期:2020-02-25
  • A two‐dimensional metric temporal logic
    Math. Logic Q. (IF 0.244) Pub Date : 2019-12-26
    Stefano Baratella; Andrea Masini

    We introduce a two‐dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut elimination theorem down to formulas of a certain type.

    更新日期:2019-12-26
  • New substitution bases for complexity classes
    Math. Logic Q. (IF 0.244) Pub Date : 2019-12-23
    Stefano Mazzanti

    The set AC 0 ( F ) , the AC 0 closure of F, is the closure with respect to substitution and concatenation recursion on notation of a set of basic functions comprehending the set F. By improving earlier work, we show that AC 0 ( F ) is the substitution closure of a simple function set and characterize well‐known function complexity classes as the substitution closure of finite sets of simple functions

    更新日期:2019-12-23
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