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Indiscernibles and satisfaction classes in arithmetic Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-28 Ali Enayat
We investigate the theory Peano Arithmetic with Indiscernibles (\(\textrm{PAI}\)). Models of \(\textrm{PAI}\) are of the form \(({\mathcal {M}},I)\), where \({\mathcal {M}}\) is a model of \(\textrm{PA}\), I is an unbounded set of order indiscernibles over \({\mathcal {M}}\), and \(({\mathcal {M}},I)\) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems
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Pcf without choice Sh835 Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-22 Saharon Shelah
We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of \(\lambda \) is well ordered for every \(\lambda \) (really local version for a given \(\lambda \)). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals
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On computable numberings of families of Turing degrees Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-18
Abstract In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show
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Around accumulation points and maximal sequences of indiscernibles Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-13 Moti Gitik
Answering a question of Mitchell (Trans Am Math Soc 329(2):507–530, 1992) we show that a limit of accumulation points can be singular in \({\mathcal {K}}\). Some additional constructions are presented.
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Varieties of truth definitions Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-09 Piotr Gruza, Mateusz Łełyk
We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence \(\alpha \) which extends a weak arithmetical theory (which we take to be \({{\,\mathrm{I\Delta _{0}+\exp }\,}}\)) such that for some formula \(\Theta \) and any arithmetical sentence \(\varphi \), \(\Theta (\ulcorner
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Essential hereditary undecidability Arch. Math. Logic (IF 0.3) Pub Date : 2024-03-01 Albert Visser
In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily
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On the extendability to $$\mathbf {\Pi }_3^0$$ ideals and Katětov order Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-26 Jialiang He, Jintao Luo, Shuguo Zhang
We show that there is a \( \varvec{\Sigma }_4^0\) ideal such that it’s neither extendable to any \( \varvec{\Pi }_3^0\) ideal nor above the ideal \( \textrm{Fin}\times \textrm{Fin} \) in the sense of Katětov order, answering a question from M. Hrušák.
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Errata: on the role of the continuum hypothesis in forcing principles for subcomplete forcing Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-19
Abstract In this note, I will list instances where in the literature on subcomplete forcing and its forcing principles (mostly in articles of my own), the assumption of the continuum hypothesis, or that we are working above the continuum, was omitted. I state the correct statements and provide or point to correct proofs. There are also some new results, most of which revolve around showing the necessity
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Vector spaces with a union of independent subspaces Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-17 Alessandro Berarducci, Marcello Mamino, Rosario Mennuni
We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete
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Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-15
Abstract Let \({\mathcal {R}}\) be a polynomially bounded o-minimal expansion of the real field. Let f(z) be a transcendental entire function of finite order \(\rho \) and type \(\sigma \in [0,\infty ]\) . The main purpose of this paper is to show that if ( \(\rho <1\) ) or ( \(\rho =1\) and \(\sigma =0\) ), the restriction of f(z) to the real axis is not definable in \({\mathcal {R}}\) . Furthermore
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The second-order version of Morley’s theorem on the number of countable models does not require large cardinals Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-14 Franklin D. Tall, Jing Zhang
The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.
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Indestructibility and the linearity of the Mitchell ordering Arch. Math. Logic (IF 0.3) Pub Date : 2024-02-13
Abstract Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \) . It then follows that \(A_0 = \{\delta < \kappa \mid \delta \) is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is nonlinear \(\}\) is unbounded in \(\kappa \) . If the Mitchell ordering of normal measures over \(\lambda \) is also linear,
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Regressive versions of Hindman’s theorem Arch. Math. Logic (IF 0.3) Pub Date : 2024-01-31 Lorenzo Carlucci, Leonardo Mainardi
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Cut elimination for coherent theories in negation normal form Arch. Math. Logic (IF 0.3) Pub Date : 2024-01-24 Paolo Maffezioli
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L-domains as locally continuous sequent calculi Arch. Math. Logic (IF 0.3) Pub Date : 2024-01-23 Longchun Wang, Qingguo Li
Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent L-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an L-domain, and every L-domain can
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Prenex normalization and the hierarchical classification of formulas Arch. Math. Logic (IF 0.3) Pub Date : 2023-12-23 Makoto Fujiwara, Taishi Kurahashi
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Weak essentially undecidable theories of concatenation, part II Arch. Math. Logic (IF 0.3) Pub Date : 2023-11-02 Juvenal Murwanashyaka
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Maximal Tukey types, P-ideals and the weak Rudin–Keisler order Arch. Math. Logic (IF 0.3) Pub Date : 2023-10-31 Konstantinos A. Beros, Paul B. Larson
In this paper, we study some new examples of ideals on \(\omega \) with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the
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On Harrop disjunction property in intermediate predicate logics Arch. Math. Logic (IF 0.3) Pub Date : 2023-10-27 Katsumasa Ishii
A partial solution to Ono’s problem P54 is given. Here Ono’s problem P54 is whether Harrop disjunction property is equivalent to disjunction property or not in intermediate predicate logics. As an application of this result it is shown that some intermediate predicate logics satisfy Harrop disjunction property.
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Stably embedded submodels of Henselian valued fields Arch. Math. Logic (IF 0.3) Pub Date : 2023-10-20 Pierre Touchard
We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued field is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields
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On diagonal functions for equivalence relations Arch. Math. Logic (IF 0.3) Pub Date : 2023-10-18 Serikzhan A. Badaev, Nikolay A. Bazhenov, Birzhan S. Kalmurzayev, Manat Mustafa
We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let E be an equivalence relation on the set of natural numbers \(\omega \), having at least two classes. A total function f is a diagonal function for E if for every x, the numbers x and f(x) are not E-equivalent.
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Compositional truth with propositional tautologies and quantifier-free correctness Arch. Math. Logic (IF 0.3) Pub Date : 2023-10-06 Bartosz Wcisło
In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is
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Effective weak and vague convergence of measures on the real line Arch. Math. Logic (IF 0.3) Pub Date : 2023-09-27 Diego A. Rojas
We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these
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Superrosiness and dense pairs of geometric structures Arch. Math. Logic (IF 0.3) Pub Date : 2023-09-19 Gareth J. Boxall
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Computable approximations of a chainable continuum with a computable endpoint Arch. Math. Logic (IF 0.3) Pub Date : 2023-09-13 Zvonko Iljazović, Matea Jelić
It is known that a semicomputable continuum S in a computable topological space can be approximated by a computable subcontinuum by any given precision under condition that S is chainable and decomposable. In this paper we show that decomposability can be replaced by the assumption that S is chainable from a to b, where a is a computable point.
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Semi-honest subrecursive degrees and the collection rule in arithmetic Arch. Math. Logic (IF 0.3) Pub Date : 2023-08-12 Andrés Cordón-Franco, F. Félix Lara-Martín
By a result of L.D. Beklemishev, the hierarchy of nested applications of the \(\Sigma _1\)-collection rule over any \(\Pi _2\)-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true \(\Pi _2\)-sentences, S
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Convergence of measures after adding a real Arch. Math. Logic (IF 0.3) Pub Date : 2023-08-11 Damian Sobota, Lyubomyr Zdomskyy
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A Mathias criterion for the Magidor iteration of Prikry forcings Arch. Math. Logic (IF 0.3) Pub Date : 2023-08-04 Omer Ben-Neria
We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.
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Herbrand complexity and the epsilon calculus with equality Arch. Math. Logic (IF 0.3) Pub Date : 2023-07-29 Kenji Miyamoto, Georg Moser
The \(\varepsilon \)-elimination method of Hilbert’s \(\varepsilon \)-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s \(\varepsilon \)-calculus focused mainly
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Revisiting the conservativity of fixpoints over intuitionistic arithmetic Arch. Math. Logic (IF 0.3) Pub Date : 2023-07-28 Mattias Granberg Olsson, Graham E. Leigh
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Turing degrees and randomness for continuous measures Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-15 Mingyang Li, Jan Reimann
We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness
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Models of $${{\textsf{ZFA}}}$$ in which every linearly ordered set can be well ordered Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-13 Paul Howard, Eleftherios Tachtsis
We provide a general criterion for Fraenkel–Mostowski models of \({\textsf{ZFA}}\) (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (\({\textsf{LW}}\)), and look at six models for \({\textsf{ZFA}}\) which satisfy this criterion (and thus \({\textsf{LW}}\) is true in these models) and “every Dedekind finite set
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Recursive Polish spaces Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-09 Tyler Arant
This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space \({\mathcal {X}}\), and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space \(\mathbb {N}\times {\mathcal {X}}\)
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On the complexity of the theory of a computably presented metric structure Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-09 Caleb Camrud, Isaac Goldbring, Timothy H. McNicholl
We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form \(\phi ^\mathcal {M}\le r\), and the open
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The fixed point and the Craig interpolation properties for sublogics of $$\textbf{IL}$$ Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-10 Sohei Iwata, Taishi Kurahashi, Yuya Okawa
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Structure of semisimple rings in reverse and computable mathematics Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-08 Huishan Wu
This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to
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A syntactic approach to Borel functions: some extensions of Louveau’s theorem Arch. Math. Logic (IF 0.3) Pub Date : 2023-06-02 Takayuki Kihara, Kenta Sasaki
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On the non-existence of $$\kappa $$ -mad families Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-23 Haim Horowitz, Saharon Shelah
Starting from a model with a Laver-indestructible supercompact cardinal \(\kappa \), we construct a model of \(ZF+DC_{\kappa }\) where there are no \(\kappa \)-mad families.
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The small index property for countable superatomic boolean algebras Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-19 J. K. Truss
It is shown that all the countable superatomic boolean algebras of finite rank have the small index property.
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An AEC framework for fields with commuting automorphisms Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-20 Tapani Hyttinen, Kaisa Kangas
In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory
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Mathias and silver forcing parametrized by density Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-18 Giorgio Laguzzi, Heike Mildenberger, Brendan Stuber-Rousselle
We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse \(2^\omega \) to \(\omega \), while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing
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Questions on cardinal invariants of Boolean algebras Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-18 Mario Jardón Santos
In the book Cardinal Invariants on Boolean Algebras by J. Donald Monk many such cardinal functions are defined and studied. Among them several are generalizations of well known cardinal characteristics of the continuum. Alongside a long list of open problems is given. Focusing on half a dozen of those cardinal invariants some of those problems are given an answer here, which in most of the cases is
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Definable Tietze extension property in o-minimal expansions of ordered groups Arch. Math. Logic (IF 0.3) Pub Date : 2023-05-15 Masato Fujita
The following two assertions are equivalent for an o-minimal expansion of an ordered group \(\mathcal M=(M,<,+,0,\ldots )\). There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function \(f:A \rightarrow M\) defined on a definable closed subset of \(M^n\) has a definable continuous extension \(F:M^n \rightarrow M\).
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Consistency and interpolation in linear continuous logic Arch. Math. Logic (IF 0.3) Pub Date : 2023-03-21 Mahya Malekghasemi, Seyed-Mohammad Bagheri
We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.
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The axiom of choice in metric measure spaces and maximal $$\delta $$ -separated sets Arch. Math. Logic (IF 0.3) Pub Date : 2023-03-13 Michał Dybowski, Przemysław Górka
We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal \(\delta \)-separated
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A topological completeness theorem for transfinite provability logic Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-22 Juan P. Aguilera
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Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-16 Iván Ongay-Valverde
In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact
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Ideals with Smital properties Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-14 Marcin Michalski, Robert Rałowski, Szymon Żeberski
A \(\sigma \)-ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set D and a Borel \(\mathcal {I}\)-positive set B the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products
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Towers, mad families, and unboundedness Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-12 Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are \({\mathcal {B}}\)-Canjar for any countably directed unbounded family \({\mathcal {B}}\) of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover
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Involutive symmetric Gödel spaces, their algebraic duals and logic Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-10 A. Di Nola, R. Grigolia, G. Vitale
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Ranks based on strong amalgamation Fraïssé classes Arch. Math. Logic (IF 0.3) Pub Date : 2023-02-02 Vincent Guingona, Miriam Parnes
In this paper, we introduce the notion of \({\textbf{K}} \)-rank, where \({\textbf{K}} \) is a strong amalgamation Fraïssé class. Roughly speaking, the \({\textbf{K}} \)-rank of a partial type is the number “copies” of \({\textbf{K}} \) that can be “independently coded” inside of the type. We study \({\textbf{K}} \)-rank for specific examples of \({\textbf{K}} \), including linear orders, equivalence
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A criterion for the strong cell decomposition property Arch. Math. Logic (IF 0.3) Pub Date : 2023-01-31 Somayyeh Tari
Let \( {\mathcal {M}}=(M, <, \ldots ) \) be a weakly o-minimal structure. Assume that \( {\mathcal {D}}ef({\mathcal {M}})\) is the collection of all definable sets of \( {\mathcal {M}} \) and for any \( m\in {\mathbb {N}} \), \( {\mathcal {D}}ef_m({\mathcal {M}}) \) is the collection of all definable subsets of \( M^m \) in \( {\mathcal {M}} \). We show that the structure \( {\mathcal {M}} \) has the
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Some implications of Ramsey Choice for families of $$\varvec{n}$$ -element sets Arch. Math. Logic (IF 0.3) Pub Date : 2022-12-16 Lorenz Halbeisen, Salome Schumacher
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Independent families and some notions of finiteness Arch. Math. Logic (IF 0.3) Pub Date : 2022-12-14 Eric Hall, Kyriakos Keremedis
In \(\textbf{ZF}\), the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of X of size \(|{\mathcal {P}} (X)|\) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product \({\textbf{2}}^{{\mathcal {P}}(X)}\) has a dense subset of size |X|”. However, the
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Glivenko sequent classes and constructive cut elimination in geometric logics Arch. Math. Logic (IF 0.3) Pub Date : 2022-12-08 Giulio Fellin, Sara Negri, Eugenio Orlandelli
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The structure of $$\kappa $$ -maximal cofinitary groups Arch. Math. Logic (IF 0.3) Pub Date : 2022-12-04 Vera Fischer, Corey Bacal Switzer
We study \(\kappa \)-maximal cofinitary groups for \(\kappa \) regular uncountable, \(\kappa = \kappa ^{<\kappa }\). Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that: (1) Any \(\kappa \)-maximal cofinitary group has \({<}\kappa \) many orbits under the natural group action of \(S(\kappa )\) on \(\kappa \). (2) If \(\mathfrak
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Generic existence of interval P-points Arch. Math. Logic (IF 0.3) Pub Date : 2022-11-07 Jialiang He, Renling Jin, Shuguo Zhang
A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\)
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Towards a homotopy domain theory Arch. Math. Logic (IF 0.3) Pub Date : 2022-11-04 Daniel O. Martínez-Rivillas, Ruy J. G. B. de Queiroz
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