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  • Tree-like constructions in topology and modal logic
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-31
    G. Bezhanishvili, N. Bezhanishvili, J. Lucero-Bryan, J. van Mill

    Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.

    更新日期:2020-07-31
  • Characterising Brouwer’s continuity by bar recursion on moduli of continuity
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-15
    Makoto Fujiwara, Tatsuji Kawai

    We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space \({{\mathbb {N}}}^{{\mathbb {N}}}\) to the natural numbers \({\mathbb {N}}\) which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction

    更新日期:2020-07-15
  • Axiomatic theory of betweenness
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-13
    Sanaz Azimipour, Pavel Naumov

    Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three sets of points. The main technical result is a sound and complete logical system describing universal properties of this relation between sets of vertices of a graph

    更新日期:2020-07-13
  • A note on uniform density in weak arithmetical theories
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-07
    Duccio Pianigiani, Andrea Sorbi

    Answering a question raised by Shavrukov and Visser (Notre Dame J Form Log 55(4):569–582, 2014), we show that the lattice of \(\exists \Sigma ^\mathsf {b}_1\)-sentences (in the language of Buss’ weak arithmetical system \(\mathsf {S}^1_2\)) over any computable enumerable consistent extension T of \(\mathsf {S}^1_2\) is uniformly dense (in the sense of Definition 2). We also show that for every \(\mathcal

    更新日期:2020-07-07
  • On countably saturated linear orders and certain class of countably saturated graphs
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-05
    Ziemowit Kostana

    The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality \(\mathfrak {c}\). We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality \(\mathfrak {c}\), under different set-theoretic assumptions

    更新日期:2020-07-05
  • Some complete $$\omega $$ ω -powers of a one-counter language, for any Borel class of finite rank
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-07-02
    Olivier Finkel, Dominique Lecomte

    We prove that, for any natural number \(n\ge 1\), we can find a finite alphabet \(\Sigma \) and a finitary language L over \(\Sigma \) accepted by a one-counter automaton, such that the \(\omega \)-power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$ is \({\varvec{\Pi }}^0_n\)-complete. We prove a similar result for the class \({\varvec{\Sigma

    更新日期:2020-07-03
  • Ring structure theorems and arithmetic comprehension
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-06-30
    Huishan Wu

    Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their \(\Pi ^{0}_{1}\) subsets and show that Schur’s Lemma is provable in \(\mathrm RCA_{0}\). A ring R

    更新日期:2020-07-01
  • Strong cell decomposition property in o-minimal traces
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-06-29
    Somayyeh Tari

    Strong cell decomposition property has been proved in non-valuational weakly o-minimal expansions of ordered groups. In this note, we show that all o-minimal traces have strong cell decomposition property. Also after introducing the notion of irrational nonvaluational cut in arbitrary o-minimal structures, we show that every expansion of o-minimal structures by irrational nonvaluational cuts is an

    更新日期:2020-06-29
  • Selection properties of the split interval and the Continuum Hypothesis
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-06-03
    Taras Banakh

    We prove that every usco multimap \(\varPhi :X\rightarrow Y\) from a metrizable separable space X to a GO-space Y has an \(F_\sigma \)-measurable selection. On the other hand, for the split interval \({\ddot{\mathbb I}}\) and the projection \(P:{{\ddot{\mathbb I}}}^2\rightarrow \mathbb I^2\) of its square onto the unit square \(\mathbb I^2\), the usco multimap \({P^{-1}:\mathbb I^2\multimap {{\ddot{\mathbb

    更新日期:2020-06-03
  • Continuous logic and embeddings of Lebesgue spaces
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-05-27
    Timothy H. McNicholl

    We use the compactness theorem of continuous logic to give a new proof that \(L^r([0,1]; {\mathbb {R}})\) isometrically embeds into \(L^p([0,1]; {\mathbb {R}})\) whenever \(1 \le p \le r \le 2\). We will also give a proof for the complex case. This will involve a new characterization of complex \(L^p\) spaces based on Banach lattices.

    更新日期:2020-05-27
  • First-order concatenation theory with bounded quantifiers
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-05-23
    Lars Kristiansen, Juvenal Murwanashyaka

    We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

    更新日期:2020-05-23
  • Easton collapses and a strongly saturated filter
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-05-17
    Masahiro Shioya

    We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses.

    更新日期:2020-05-17
  • Logics of left variable inclusion and Płonka sums of matrices
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-04-13
    S. Bonzio, T. Moraschini, M. Pra Baldi

    The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \(\vdash \). We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \(\vdash \) is related to the construction of Płonka sums of the matrix models of \(\vdash \). This observation allows to obtain a Hilbert-style axiomatization

    更新日期:2020-04-13
  • Strong downward Löwenheim–Skolem theorems for stationary logics, I
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-04-09
    Sakaé Fuchino, André Ottenbreit Maschio Rodrigues, Hiroshi Sakai

    This note concerns the model theoretic properties of logics extending the first-order logic with monadic (weak) second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem (SDLS) down to \(<\aleph _2\) for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into

    更新日期:2020-04-09
  • $$AD_{\mathbb {R}}$$ADR implies that all sets of reals are $$\Theta $$Θ universally Baire
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-04-08
    Grigor Sargsyan

    We show that assuming the determinacy of all games on reals, every set of reals is \(\Theta \) universally baire.

    更新日期:2020-04-08
  • Special ultrafilters and cofinal subsets of $$({}^\omega \omega , <^*)$$(ωω,<∗)
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-04-05
    Peter Nyikos

    The interplay between ultrafilters and unbounded subsets of \({}^\omega \omega \) with the order \(<^*\) of strict eventual domination is studied. Among the tools are special kinds of non-principal (“free”) ultrafilters on \(\omega \). These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \(\subset ^*\) of almost inclusion. It

    更新日期:2020-04-05
  • Fields with automorphism and valuation
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-31
    Özlem Beyarslan, Daniel Max Hoffmann, Gönenç Onay, David Pierce

    The model companion of the theory of fields with valuation and automorphism (of the pure field structure) exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion.

    更新日期:2020-03-31
  • Ideal generalizations of Egoroff’s theorem
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-30
    Miroslav Repický

    We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch (Real Anal Exchange 42(2):269–282, 2017).

    更新日期:2020-03-30
  • A diamond-plus principle consistent with AD
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-04
    Daniel W. Cunningham

    After showing that \(AD +DC \) refutes \(\lozenge ^+_\kappa \) for all regular cardinals \(\kappa \ge \omega _1\), we present a diamond-plus principle \(\lozenge _{{{\mathbb {R}}} }^+\) concerning all subsets of \(\varTheta \). Using a forcing argument, we prove that \(\lozenge _{{{\mathbb {R}}} }^+\) holds in Steel’s core model \({{{{\mathbf {K}}}({{{{\mathbb {R}}} }})}}\), an inner model in which

    更新日期:2020-02-04
  • Scattered sentences have few separable randomizations
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-03
    Uri Andrews, Isaac Goldbring, Sherwood Hachtman, H. Jerome Keisler, David Marker

    In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \(L_{\omega _1\omega }\)-sentence with few separable

    更新日期:2020-02-03
  • Some nondefinability results with entire functions in a polynomially bounded o-minimal structure
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-31
    Hassan Sfouli

    Let \(f(z)=\Sigma _{k\ge 0}a_{k}z^{k}\) be a transcendental entire function with real coefficients. The main purpose of this paper is to show that the restriction of f to \(\mathbb {R}\) is not definable in the ordered field of real numbers with restricted analytic functions, \(\mathbb {R}_{\text {an}}\). Furthermore, we show that there is \(\theta \in \mathbb {R}\) such that the function \(f(xe^{i\theta

    更新日期:2020-01-31
  • Completeness of the primitive recursive $$\omega $$ ω -rule
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-30
    Emanuele Frittaion

    Shoenfield’s completeness theorem (1959) states that every true first order arithmetical sentence has a recursive \(\omega \)-proof encodable by using recursive applications of the \(\omega \)-rule. For a suitable encoding of Gentzen style \(\omega \)-proofs, we show that Shoenfield’s completeness theorem applies to cut free \(\omega \)-proofs encodable by using primitive recursive applications of

    更新日期:2020-01-30
  • Definable combinatorics with dense linear orders
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-22
    Himanshu Shukla, Arihant Jain, Amit Kuber

    We compute the model-theoretic Grothendieck ring, \(K_0({\mathcal {Q}})\), of a dense linear order (DLO) with or without end points, \({\mathcal {Q}}=(Q,<)\), as a structure of the signature \(\{<\}\), and show that it is a quotient of the polynomial ring over \({\mathbb {Z}}\) generated by \({\mathbb {N}}_+\times (Q\sqcup \{-\infty \})\) by an ideal that encodes multiplicative relations of pairs of

    更新日期:2020-01-22
  • The noneffectivity of Arslanov’s completeness criterion and related theorems
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-22
    Sebastiaan A. Terwijn

    We discuss the (non)effectivity of Arslanov’s completeness criterion. In particular, we show that a parameterized version, similar to the recursion theorem with parameters, fails. We also discuss the effectivity of another extension of the recursion theorem, namely Visser’s ADN theorem, as well as that of a joint generalization of the ADN theorem and Arslanov’s completeness criterion.

    更新日期:2020-01-22
  • Uniform Lyndon interpolation property in propositional modal logics
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-21
    Taishi Kurahashi

    We introduce and investigate the notion of uniform Lyndon interpolation property (ULIP) which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \(\mathbf{K}\), \(\mathbf{KB}\), \(\mathbf{GL}\) and \(\mathbf{Grz}\) enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property

    更新日期:2020-01-21
  • Dependent choice as a termination principle
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-16
    Thomas Powell

    We introduce a new formulation of the axiom of dependent choice, which can be viewed as an abstract termination principle that in particular generalises recursive path orderings, the latter being fundamental tools used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

    更新日期:2020-01-16
  • Finite sets and infinite sets in weak intuitionistic arithmetic
    Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-02
    Takako Nemoto

    In this paper, we consider, for a set \(\mathcal {A}\) of natural numbers, the following notions of finiteness FIN1: There are a natural number l and a bijection f between \(\{ x\in \mathbb {N}:xy)(x\in \mathcal {A})\); FIN5: It is not the case that \(\forall l\exists \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|=l)\), and infiniteness INF1: There are not a natural number l and a bijection f between

    更新日期:2020-01-02
  • On Ramsey choice and partial choice for infinite families of n -element sets
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-12-06
    Lorenz Halbeisen, Eleftherios Tachtsis

    For an integer \(n\ge 2\), Ramsey Choice\(\mathsf {RC}_{n}\) is the weak choice principle “every infinite setxhas an infinite subset y such that\([y]^{n}\) (the set of alln-element subsets of y) has a choice function”, and \(\mathsf {C}_{n}^{-}\) is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for

    更新日期:2019-12-06
  • Reversibility of extreme relational structures
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-11-27
    Miloš S. Kurilić, Nenad Morača

    A relational structure \({{\mathbb {X}}}\) is called reversible iff each bijective homomorphism from \({{\mathbb {X}}}\) onto \({{\mathbb {X}}}\) is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations

    更新日期:2019-11-27
  • Rank-initial embeddings of non-standard models of set theory
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-11-14
    Paul Kindvall Gorbow

    A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are

    更新日期:2019-11-14
  • Induction rules in bounded arithmetic
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-11-12
    Emil Jeřábek

    We study variants of Buss’s theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on \(\hat{\varPi }^{b}_i\) induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for \(T^i_2\)

    更新日期:2019-11-12
  • Scott sentences for equivalence structures
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-11-08
    Sara B. Quinn

    For a computable structure \({\mathcal {A}}\), if there is a computable infinitary Scott sentence, then the complexity of this sentence gives an upper bound for the complexity of the index set \(I({\mathcal {A}})\). If we can also show that \(I({\mathcal {A}})\) is m-complete at that level, then there is a correspondence between the complexity of the index set and the complexity of a Scott sentence

    更新日期:2019-11-08
  • Covering properties of $$\omega $$ω -mad families
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-11-08
    Leandro Aurichi, Lyubomyr Zdomskyy

    We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \(\mathfrak b=\mathfrak c\) is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

    更新日期:2019-11-08
  • Analytic computable structure theory and $$L^p$$Lp -spaces part 2
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-10-31
    Tyler Brown, Timothy H. McNicholl

    Suppose \(p \ge 1\) is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \(L^p\) spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.

    更新日期:2019-10-31
  • Square below a non-weakly compact cardinal
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-10-25
    Hazel Brickhill

    In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under \(V=L\) any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very

    更新日期:2019-10-25
  • Absorbing the structural rules in the sequent calculus with additional atomic rules
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-10-24
    Franco Parlamento, Flavio Previale

    We show that if the structural rules are admissible over a set \(\mathcal{R}\) of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in \(\mathcal{R}\) to the multisuccedent minimal and intuitionistic \(\mathbf{G3}\) calculi as well as to the classical one. Two applications to pure logic and to the sequent calculus with equality are presented.

    更新日期:2019-10-24
  • Antichains of perfect and splitting trees
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-09-20
    Paul Hein, Otmar Spinas

    We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \(\mathsf {cov}(\mathcal {M})\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are

    更新日期:2019-09-20
  • Weaker variants of infinite time Turing machines
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-09-13
    Matteo Bianchetti

    Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the \(\limsup \) of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the \(\limsup \) rule with an ‘eventually constant’ rule: at each

    更新日期:2019-09-13
  • A small ultrafilter number at smaller cardinals
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-09-10
    Dilip Raghavan, Saharon Shelah

    It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on \({\aleph }_{\omega +1}\) which is generated by fewer than \({2}^{{\aleph }_{\omega +1}}\) sets.

    更新日期:2019-09-10
  • On the forking topology of a reduct of a simple theory
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-08-29
    Ziv Shami

    Let T be a simple L-theory and let \(T^-\) be a reduct of T to a sublanguage \(L^-\) of L. For variables x, we call an \(\emptyset \)-invariant set \(\Gamma (x)\) in \({\mathcal {C}}\) a universal transducer if for every formula \(\phi ^-(x,y)\in L^-\) and every a, $$\begin{aligned} \phi ^-(x,a)\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma (x)\wedge \phi ^-(x,a)\ L\text{-forks

    更新日期:2019-08-29
  • Detecting properties from descriptions of groups
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-08-17
    Iva Bilanovic, Jennifer Chubb, Sam Roven

    We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic

    更新日期:2019-08-17
  • Ordinal analyses for monotone and cofinal transfinite inductions
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-08-14
    Kentaro Sato

    We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \(\varDelta _0\)

    更新日期:2019-08-14
  • Proof-theoretic strengths of the well-ordering principles
    Arch. Math. Logic (IF 0.485) Pub Date : 2019-07-30
    Toshiyasu Arai

    In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions \(\mathsf {g}\) on ordinals is shown to be equal to the least fixed point of \(\mathsf {g}\). Moreover corrections to the previous paper (Arai in Arch Math Log 57:649–664, 2017) are made.

    更新日期:2019-07-30
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