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A property of forcing notions and preservation of cardinal invariants Math. Logic Q. (IF 0.3) Pub Date : 2023-12-04 Yushiro Aoki
We define a property of forcing notions and show that there exists a model of its forcing axiom and the negation of the continuum hypothesis in which the Cichoń-Blass diagram of cardinal invariants is the same as in the Cohen model. As a corollary, its forcing axiom and the forcing axiom for σ-centered forcing notions are independent of each other.
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Infinitary logic with infinite sequents: syntactic investigations Math. Logic Q. (IF 0.3) Pub Date : 2023-12-04 Matteo Tesi
The present paper deals with a purely syntactic analysis of infinitary logic with infinite sequents. In particular, we discuss sequent calculi for classical and intuitionistic infinitary logic with good structural properties based on sequents possibly containing infinitely many formulas. A cut admissibility proof is proposed which employs a new strategy and a new inductive parameter. We conclude the
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A dichotomy for T-convex fields with a monomial group Math. Logic Q. (IF 0.3) Pub Date : 2023-11-21 Elliot Kaplan, Christoph Kesting
We prove a dichotomy for o-minimal fields R $\mathcal {R}$ , expanded by a T-convex valuation ring (where T is the theory of R $\mathcal {R}$ ) and a compatible monomial group. We show that if T is power bounded, then this expansion of R $\mathcal {R}$ is model complete (assuming that T is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if R $\mathcal {R}$
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On bQ1$bQ_1$-degrees of c.e. sets Math. Logic Q. (IF 0.3) Pub Date : 2023-11-20 Roland Omanadze, Irakli Chitaia
Using properties of simple sets we study b Q 1 ${bQ}_1$ -degrees of c.e. sets. In particular, we prove: (1) If A and B are c.e. sets, A is a simple set and A ≤ b Q 1 B $A\le _{{bQ}_{1}}B$ , then there exists a simple set C such that C ≤ 1 A $C\le _1 A$ and C ≤ 1 B $C\le _1 B$ . (2) the c.e. b Q 1 ${bQ}_1$ -degrees ( b Q 1 ${bQ}_1$ -degrees) do not form an upper semilattice. (3) The c.e. b Q 1 ${bQ}_1$
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The persistence principle over weak interpretability logic Math. Logic Q. (IF 0.3) Pub Date : 2023-10-27 Sohei Iwata, Taishi Kurahashi, Yuya Okawa
We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut-elimination theorem for it. As a consequence
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Infinite Wordle and the mastermind numbers Math. Logic Q. (IF 0.3) Pub Date : 2023-09-13 Joel David Hamkins
I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback
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A classification of low c.e. sets and the Ershov hierarchy Math. Logic Q. (IF 0.3) Pub Date : 2023-09-11 Marat Faizrahmanov
In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ-levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ-level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability
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When cardinals determine the power set: inner models and Härtig quantifier logic Math. Logic Q. (IF 0.3) Pub Date : 2023-09-11 Jouko Väänänen, Philip D. Welch
We show that the predicate “x is the power set of y” is Σ 1 ( Card ) $\Sigma _1(\operatorname{Card})$ -definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here Card $\operatorname{Card}$ is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic
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Approximate isomorphism of metric structures Math. Logic Q. (IF 0.3) Pub Date : 2023-09-05 James E. Hanson
We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our
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Pregeometry over locally o-minimal structures and dimension Math. Logic Q. (IF 0.3) Pub Date : 2023-08-30 Masato Fujita
We define a discrete closure operator for definably complete locally o-minimal structures M $\mathcal {M}$ . The pair of the underlying set of M $\mathcal {M}$ and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it discl $\operatorname{discl}$ -dimension. A definable set X is of dimension equal to the discl
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Bisimulations and bisimulation games between Verbrugge models Math. Logic Q. (IF 0.3) Pub Date : 2023-08-04 Sebastijan Horvat, Tin Perkov, Mladen Vuković
Interpretability logic is a modal formalization of relative interpretability between first-order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call
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On a cardinal inequality in ZF$\mathsf {ZF}$ Math. Logic Q. (IF 0.3) Pub Date : 2023-08-04 Guozhen Shen
It is proved in ZF $\mathsf {ZF}$ (without the axiom of choice) that a n ⩽ S n + 1 ( a ) $\mathfrak {a}^n\leqslant \mathcal {S}_{n+1}(\mathfrak {a})$ for all infinite cardinals a $\mathfrak {a}$ and all natural numbers n ≠ 0 $n\ne 0$ , where S n + 1 ( a ) $\mathcal {S}_{n+1}(\mathfrak {a})$ is the cardinality of the set of permutations with exactly n + 1 $n+1$ non-fixed points of a set which is of
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The permutations with n non-fixed points and the subsets with n elements of a set Math. Logic Q. (IF 0.3) Pub Date : 2023-08-04 Supakun Panasawatwong, Pimpen Vejjajiva
We write S n ( a ) $\mathcal {S}_n(\mathfrak {a})$ and [ a ] n $[\mathfrak {a}]^n$ for the cardinalities of the set of permutations with n non-fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality a $\mathfrak {a}$ , where n is a natural number greater than 1. With the Axiom of Choice, S n ( a ) $\mathcal {S}_n(\mathfrak {a})$ and [ a ] n $[\mathfrak {a}]^n$
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On self-distributive weak Heyting algebras Math. Logic Q. (IF 0.3) Pub Date : 2023-08-02 Mohsen Nourany, Shokoofeh Ghorbani, Arsham Borumand Saeid
We use the left self-distributive axiom to introduce and study a special class of weak Heyting algebras, called self-distributive weak Heyting algebras (SDWH-algebras). We present some useful properties of SDWH-algebras and obtain some equivalent conditions of them. A characteristic of SDWH-algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH-algebras and
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Models of VTC0$\mathsf {VTC^0}$ as exponential integer parts Math. Logic Q. (IF 0.3) Pub Date : 2023-08-02 Emil Jeřábek
We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory VTC 0 $\mathsf {VTC^0}$ are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically bounded numbers. Combined with our previous results on the construction of the real exponential function on completions of models of VTC 0 $\mathsf {VTC^0}$
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Borel complexity and Ramsey largeness of sets of oracles separating complexity classes Math. Logic Q. (IF 0.3) Pub Date : 2023-08-02 Alex Creiner, Stephen Jackson
We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, P ≠ NP $\mathbf {P}\ne \mathbf {NP}$ with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that
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Forcing revisited Math. Logic Q. (IF 0.3) Pub Date : 2023-08-01 Toby Meadows
The purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model
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Logics of upsets of De Morgan lattices Math. Logic Q. (IF 0.3) Pub Date : 2023-07-31 Adam Přenosil
We study logics determined by matrices consisting of a De Morgan lattice with an upward closed set of designated values, such as the logic of non-falsity preservation in a given finite Boolean algebra and Shramko's logic of non-falsity preservation in the four-element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice-theoretic notion of an n-filter
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Spherically complete models of Hensel minimal valued fields Math. Logic Q. (IF 0.3) Pub Date : 2023-07-28 David B. Bradley-Williams, Immanuel Halupczok
We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0-hmix-minimality (which, in equi-characteristic 0, amounts to 0-h-minimality).
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Coding of real-valued continuous functions under WKL$\mathsf {WKL}$ Math. Logic Q. (IF 0.3) Pub Date : 2023-07-24 Tatsuji Kawai
In the context of constructive reverse mathematics, we show that weak Kőnig's lemma ( WKL $\mathsf {WKL}$ ) implies that every pointwise continuous function f : [ 0 , 1 ] → R $f : [0,1]\rightarrow \mathbb {R}$ is induced by a code in the sense of reverse mathematics. This, combined with the fact that WKL $\mathsf {WKL}$ implies the Fan theorem, shows that WKL $\mathsf {WKL}$ implies the uniform continuity
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Avoiding Medvedev reductions inside a linear order Math. Logic Q. (IF 0.3) Pub Date : 2023-07-24 Noah Schweber
While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik-reducible to J itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct
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Topological duality for orthomodular lattices Math. Logic Q. (IF 0.3) Pub Date : 2023-07-24 Joseph McDonald, Katalin Bimbó
A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices
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On Hausdorff operators in ZF$\mathsf {ZF}$ Math. Logic Q. (IF 0.3) Pub Date : 2023-07-24 Kyriakos Keremedis, Eleftherios Tachtsis
A Hausdorff space ( X , T ) $(X,\mathcal {T})$ is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every x , y ∈ X $x,y\in X$ with x ≠ y $x\ne y$ , F ( x , y ) = ( U , V ) $F(x,y)=(U,V)$ , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in ZF $\mathsf {ZF}$ , i.e., in Zermelo–Fraenkel
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Bowtie-free graphs and generic automorphisms Math. Logic Q. (IF 0.3) Pub Date : 2023-07-24 Daoud Siniora
We show that the countable universal ω-categorical bowtie-free graph admits generic automorphisms. Moreover, we show that this graph is not finitely homogenisable.
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A categorical equivalence between logical quantale modules and quantum B-modules Math. Logic Q. (IF 0.3) Pub Date : 2023-07-20 Xianglong Ruan, Xiaochuan Liu
This paper introduces the notion of logical quantale module. It proves that there is a dual equivalence between the category of logical quantale modules and the category of quantum B-modules, in the way that every quantum B-module admits a natural embedding into a logical quantale module, the enveloping quantale module.
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Topological properties of definable sets in ordered Abelian groups of burden 2 Math. Logic Q. (IF 0.3) Pub Date : 2023-07-14 Alfred Dolich, John Goodrick
We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle-third set (Theorem 2
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On the variety of strong subresiduated lattices Math. Logic Q. (IF 0.3) Pub Date : 2023-07-11 Sergio Celani, Hernán J. San Martín
A subresiduated lattice is a pair ( A , D ) $(A,D)$ , where A is a bounded distributive lattice, D is a bounded sublattice of A and for every a , b ∈ A $a,b\in A$ there exists the maximum of the set { d ∈ D : a ∧ d ≤ b } $\lbrace d\in D:a\wedge d\le b\rbrace$ , which is denoted by a → b $a\rightarrow b$ . This pair can be regarded as an algebra ( A , ∧ , ∨ , → , 0 , 1 ) $(A,\wedge ,\vee ,\rightarrow
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Math. Logic Q. (IF 0.3) Pub Date : 2023-06-01
Dear Readers, We wish all of you a happy and successful year 2023. The issue you are looking at presents our journal in a new layout, the new standardised journal design used by our publisher Wiley-VCH Verlag. Apart from the major changes on the title page of articles, most of the features of the layout of the journal Mathematical Logic Quarterly (MLQ) were retained. But the appearance of the papers
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Cofinal types on ω2 Math. Logic Q. (IF 0.3) Pub Date : 2023-05-31 Borisa Kuzeljevic, Stevo Todorcevic
In this paper we start the analysis of the class D ℵ 2 $\mathcal {D}_{\aleph _2}$ , the class of cofinal types of directed sets of cofinality at most ℵ2. We compare elements of D ℵ 2 $\mathcal {D}_{\aleph _2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in D ℵ 2 $\mathcal {D}_{\aleph _2}$ , and then proceed to find some of these types which have an immediate successor
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Nice ℵ1 generated non-P-points, Part I Math. Logic Q. (IF 0.3) Pub Date : 2023-05-29 Saharon Shelah
We define a family of non-principal ultrafilters on N ${\mathbb {N}}$ which are, in a sense, very far from P-points. We prove the existence of such ultrafilters under reasonable conditions. In subsequent articles, we intend to prove that such ultrafilters may exist while no P-point exists. Though our primary motivations came from forcing and independence results, the family of ultrafilters introduced
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The cofinality of the strong measure zero ideal for κ inaccessible Math. Logic Q. (IF 0.3) Pub Date : 2023-05-29 Johannes Philipp Schürz
We investigate the cofinality of the strong measure zero ideal for κ inaccessible and show that it is independent of the size of 2κ.
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Some definable types that cannot be amalgamated Math. Logic Q. (IF 0.3) Pub Date : 2023-05-29 Martin Hils, Rosario Mennuni
We exhibit a theory where definable types lack the amalgamation property.
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A proof-theoretic metatheorem for tracial von Neumann algebras Math. Logic Q. (IF 0.3) Pub Date : 2023-05-29 Liviu Păunescu, Andrei Sipoş
We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research programme that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.
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The subset relation and 2-stratified sentences in set theory and class theory Math. Logic Q. (IF 0.3) Pub Date : 2023-05-28 Zachiri McKenzie
Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of ZF $\mathsf {ZF}$ , BAS $\mathsf {BAS}$ , that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised
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On splitting trees Math. Logic Q. (IF 0.3) Pub Date : 2023-05-28 Giorgio Laguzzi, Heike Mildenberger, Brendan Stuber-Rousselle
We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any absolute amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Hein's and Spinas' conjecture that
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Incomparable Vγ$V_\gamma$-degrees Math. Logic Q. (IF 0.3) Pub Date : 2023-05-26 Teng Zhang
In [3], Shi proved that there exist incomparable Zermelo degrees at γ if there exists an ω-sequence of measurable cardinals, whose limit is γ. He asked whether there is a size γ ω $\gamma ^\omega$ antichain of Zermelo degrees. We consider this question for the V γ $V_\gamma$ -degree structure. We use a kind of Prikry-type forcing to show that if there is an ω-sequence of measurable cardinals, then
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A note on fsg$\text{fsg}$ groups in p-adically closed fields Math. Logic Q. (IF 0.3) Pub Date : 2023-05-24 Will Johnson
Let G be a definable group in a p-adically closed field M. We show that G has finitely satisfiable generics ( fsg $\text{fsg}$ ) if and only if G is definably compact. The case M = Q p $M = \mathbb {Q}_p$ was previously proved by Onshuus and Pillay.
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Strongly unfoldable, splitting and bounding Math. Logic Q. (IF 0.3) Pub Date : 2023-05-24 Ömer Faruk Bağ, Vera Fischer
Assuming GCH $\mathsf {GCH}$ , we show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under a one step iteration of the Hechler forcing for adding a dominating κ-real. Moreover, we show that if κ is strongly unfoldable, 2 κ = κ + $2^\kappa =\kappa ^+$ and λ is a regular cardinal such that κ + < λ $\kappa ^+ < \lambda$ , then there is a set generic extension
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Decomposition into special submanifolds Math. Logic Q. (IF 0.3) Pub Date : 2023-05-10 Masato Fujita
We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.
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The power set and the set of permutations with finitely many non-fixed points of a set Math. Logic Q. (IF 0.3) Pub Date : 2023-05-10 Guozhen Shen
For a cardinal a $\mathfrak {a}$ , we write S fin ( a ) $\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for the cardinality of the set of permutations with finitely many non-fixed points of a set which is of cardinality a $\mathfrak {a}$ . We investigate the relationships between 2 a $2^\mathfrak {a}$ and S fin ( a ) $\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for an arbitrary
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Piece selection and cardinal arithmetic Math. Logic Q. (IF 0.3) Pub Date : 2022-09-07 Pierre Matet
We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if λ≥2κ$\lambda \ge 2^\kappa$, then (a) Iκ,λ$I_{\kappa , \lambda }$ is not (λ, 2)-distributive, and (b) Iκ,λ+→(Iκ,λ+)ω2$I_{\kappa , \lambda }^+ \rightarrow (I_{\kappa , \lambda }^+)^2_\omega$ does not hold.
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Intuitionistic propositional probability logic Math. Logic Q. (IF 0.3) Pub Date : 2022-08-29 Anelina Ilić-Stepić, Mateja Knežević, Zoran Ognjanović
We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and
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Automorphism invariant measures and weakly generic automorphisms Math. Logic Q. (IF 0.3) Pub Date : 2022-08-27 Gábor Sági
Let A$\mathcal {A}$ be a countable ℵ0-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group Aut(A)$\operatorname{Aut}(\mathcal {A})$ of A$\mathcal {A}$; the secondary motivation is to study the existence of weakly generic automorphisms of A$\mathcal {A}$. Among others, we present sufficient conditions implying
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A note on edge colorings and trees Math. Logic Q. (IF 0.3) Pub Date : 2022-08-20 Adi Jarden, Ziv Shami
We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies μ+<κ$\mu ^+<\kappa$. Another result is that an uncountable cardinal κ is
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Tameness of definably complete locally o-minimal structures and definable bounded multiplication Math. Logic Q. (IF 0.3) Pub Date : 2022-08-20 Masato Fujita, Tomohiro Kawakami, Wataru Komine
We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal
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Extremal numberings and fixed point theorems Math. Logic Q. (IF 0.3) Pub Date : 2022-07-20 Marat Faizrahmanov
We consider so-called extremal numberings that form the greatest or minimal degrees under the reducibility of all A-computable numberings of a given family of subsets of N$\mathbb {N}$, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A-computable numberings, respectively. The main question of this paper is when a universal
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Algebraic completion without the axiom of choice Math. Logic Q. (IF 0.3) Pub Date : 2022-07-08 Jørgen Harmse
Läuchli and Pincus showed that existence of algebraic completions of all fields cannot be proved from Zermelo-Fraenkel set theory alone. On the other hand, important special cases do follow. In particular, I show that an algebraic completion of Qp$\mathbb {Q}_p$ can be constructed in Zermelo-Fraenkel set theory.
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Theory and application of labelling techniques for interpretability logics Math. Logic Q. (IF 0.3) Pub Date : 2022-07-07 Evan Goris, Marta Bílková, Joost J. Joosten, Luka Mikec
The notion of a critical successor [5] in relational semantics has been central to most classic modal completeness proofs in interpretability logics. In this paper we shall work with a more general notion, that of an assuring successor. This will enable more concisely formulated completeness proofs, both with respect to ordinary and generalised Veltman semantics. Due to their interesting theoretical
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Choiceless large cardinals and set-theoretic potentialism Math. Logic Q. (IF 0.3) Pub Date : 2022-07-06 Raffaella Cutolo, Joel David Hamkins
We define a potentialist system of ZF$\mathsf {ZF}$-structures, i.e., a collection of possible worlds in the language of ZF$\mathsf {ZF}$ connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact,
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On the logic of distributive nearlattices Math. Logic Q. (IF 0.3) Pub Date : 2022-07-01 Luciano J. González
We study the propositional logic SDNSDN$\mathcal {S}_\mathbb {DN}$ associated with the variety of distributive nearlattices DNDN$\mathbb {DN}$ . We prove that the logic SDNSDN$\mathcal {S}_\mathbb {DN}$ coincides with the assertional logic associated with the variety DNDN$\mathbb {DN}$ and with the order-based logic associated with DNDN$\mathbb {DN}$ . We obtain a characterization of the reduced matrix
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κ-Madness and definability Math. Logic Q. (IF 0.3) Pub Date : 2022-06-21 Haim Horowitz, Saharon Shelah
Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal κ, there are no Σ11(κ)Σ11(κ)$\Sigma ^1_1(\kappa )$ κ-mad families.
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Refining the arithmetical hierarchy of classical principles Math. Logic Q. (IF 0.3) Pub Date : 2022-06-09 Makoto Fujiwara, Taishi Kurahashi
We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.
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Choice principles in local mantles Math. Logic Q. (IF 0.3) Pub Date : 2022-05-07 Farmer Schlutzenberg
Assume ZFC $\mathsf {ZFC}$ ZFC . Let κ be a cardinal. A < κ ${\mathord {<}\hspace{1.111pt}\kappa }$ <κ -ground is a transitive proper class W modelling ZFC $\mathsf {ZFC}$ ZFC such that V is a generic extension of W via a forcing P ∈ W $\mathbb {P}\in W$ of cardinality < κ ${\mathord {<}\hspace{1.111pt}\kappa }$ . The κ-mantle M κ $\mathcal {M}_\kappa$ is the intersection of all < κ ${\mathord {<}\hspace{1
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Controlling the number of normal measures at successor cardinals Math. Logic Q. (IF 0.3) Pub Date : 2022-04-28 Arthur W. Apter
We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom UAUA$\mathsf {UA}$ (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish
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Some model theory of Th(N,·)$\operatorname{Th}(\mathbb {N},\cdot )$ Math. Logic Q. (IF 0.3) Pub Date : 2022-04-28 Atticus Stonestrom
‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid ( N , · ) $(\mathbb {N},\cdot )$ . We give a full characterization of the ⌀ $\varnothing$ -definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries
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Determinacy and regularity properties for idealized forcings Math. Logic Q. (IF 0.3) Pub Date : 2022-04-27 Daisuke Ikegami
We show under ZF + DC + AD R $\sf {ZF}+ \sf {DC}+ \sf {AD}_\mathbb {R}$ that every set of reals is I-regular for any σ-ideal I on the Baire space ω ω $\omega ^{\omega }$ such that P I $\mathbb {P}_I$ is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under ZF + DC + AD + $\sf {ZF}+ \sf {DC}+ \sf {AD}^+$ if we additionally assume that the
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Gap-2 morass-definable η1-orderings Math. Logic Q. (IF 0.3) Pub Date : 2022-04-12 Bob A. Dumas
We prove that in the Cohen extension adding ℵ3 generic reals to a model of 𝖹𝖥𝖢+𝖢𝖧ZFC+CH$\mathsf {ZFC}+\mathsf {CH}$ containing a simplified (ω1, 2)-morass, gap-2 morass-definable η1-orderings with cardinality ℵ3 are order-isomorphic. Hence it is consistent that 2ℵ0=ℵ32ℵ0=ℵ3$2^{\aleph _0}=\aleph _3$ and that morass-definable η1-orderings with cardinality of the continuum are order-isomorphic. We
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The theory of hereditarily bounded sets Math. Logic Q. (IF 0.3) Pub Date : 2022-04-03 Emil Jeřábek
We show that for any 𝑘∈𝜔k∈ω$k\in \omega$ , the structure ⟨𝐻𝑘,∈⟩⟨Hk,∈⟩$\langle H_k,{\in }\rangle$ of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure 𝑉𝜔=⋃𝑘𝐻𝑘Vω=⋃kHk$V_\omega =\bigcup _kH_k$
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Bounding 2d functions by products of 1d functions Math. Logic Q. (IF 0.3) Pub Date : 2022-03-30 François Dorais, Dan Hathaway
Given sets X , Y $X,Y$ and a regular cardinal μ, let Φ ( X , Y , μ ) $\Phi (X,Y,\mu )$ be the statement that for any function f : X × Y → μ $f : X \times Y \rightarrow \mu$ , there are functions g 1 : X → μ $g_1 : X \rightarrow \mu$ and g 2 : Y → μ $g_2 : Y \rightarrow \mu$ such that for all ( x , y ) ∈ X × Y $(x,y) \in X \times Y$ , f ( x , y ) ≤ max { g 1 ( x ) , g 2 ( y ) } $f(x,y) \le \max \lbrace
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Rogers semilattices of limitwise monotonic numberings Math. Logic Q. (IF 0.3) Pub Date : 2022-03-22 Nikolay Bazhenov, Manat Mustafa, Zhansaya Tleuliyeva
Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S ⊂ P ( ω ) $S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set ν ( k ) $\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice R