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Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-06 Sakaé Fuchino, André Ottenbreit Maschio Rodrigues, Hiroshi Sakai
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Forcing the Mapping Reflection Principle by finite approximations Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-06 Tadatoshi Miyamoto, Teruyuki Yorioka
Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is \(\aleph _2\). The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the
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Model completeness and relative decidability Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-03 Jennifer Chubb, Russell Miller, Reed Solomon
We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model \(\mathcal {A}\) of a computably enumerable, model complete theory, the entire elementary diagram \(E(\mathcal {A})\) must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a uniform procedure
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Towers and clubs Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-02 Pierre Matet
We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal J extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal \(\kappa \), our goal being to witness the nonsaturation of J by the existence of towers (of length
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Sofic profiles of $$S(\omega )$$ S ( ω ) and computability Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-01 Aleksander Ivanov
We show that for every sofic chunk E there is a bijective homomorphism \(f:E_c \rightarrow E\), where \(E_c\) is a chunk of the group of computable permutations of \(\mathbb {N}\) so that the approximating morphisms of E can be viewed as restrictions of permutations of \(E_c\) to finite subsets of \(\mathbb {N}\). Using this we study some relevant effectivity conditions associated with sofic chunks
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Another method for constructing models of not approachability and not SCH Arch. Math. Logic (IF 0.485) Pub Date : 2021-01-01 Moti Gitik
We present a new method of constructing a model of \(\lnot \)SCH+\(\lnot \)AP.
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The absorption law Arch. Math. Logic (IF 0.485) Pub Date : 2020-12-16 Albert Visser
In this paper, we show how to construct for a given consistent theory U a \(\varSigma ^0_1\)-predicate that both satisfies the Löb Conditions and the Kreisel Condition—even if U is unsound. We do this in such a way that U itself can verify satisfaction of an internal version of the Kreisel Condition.
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Keisler’s order via Boolean ultrapowers Arch. Math. Logic (IF 0.485) Pub Date : 2020-09-27 Francesco Parente
In this paper, we provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of ‘separation of variables’ recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in
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A fixed-point theorem for definably amenable groups Arch. Math. Logic (IF 0.485) Pub Date : 2020-09-20 Juan Felipe Carmona, Kevin Dávila, Alf Onshuus, Rafael Zamora
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
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Cichoń’s diagram and localisation cardinals Arch. Math. Logic (IF 0.485) Pub Date : 2020-09-15 Martin Goldstern, Lukas Daniel Klausner
We reimplement the creature forcing construction used by Fischer et al. (Arch Math Log 56(7–8):1045–1103, 2017. https://doi.org/10.1007/S00153-017-0553-8. arXiv:1402.0367 [math.LO]) to separate Cichoń’s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction by adding uncountably many additional cardinal characteristics
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Quantum logic is undecidable Arch. Math. Logic (IF 0.485) Pub Date : 2020-09-11 Tobias Fritz
We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature \((\vee ,\perp ,0,1)\), where ‘\(\perp \)’ is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent
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Small sets in Mann pairs Arch. Math. Logic (IF 0.485) Pub Date : 2020-09-02 Pantelis E. Eleftheriou
Let \(\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle \) be an expansion of a real closed field \({{{\mathcal {M}}}}\) by a dense subgroup G of \(\langle M^{>0}, \cdot \rangle \) with the Mann property. We prove that the induced structure on G by \({{{\mathcal {M}}}}\) eliminates imaginaries. As a consequence, every small set X definable in \({{{\mathcal {M}}}}\) can be definably embedded
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Learning theory in the arithmetic hierarchy II Arch. Math. Logic (IF 0.485) Pub Date : 2020-08-26 Achilles A. Beros, Konstantinos A. Beros, Daniel Flores, Umar Gaffar, David J. Webb, Soowhan Yoon
The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex\(^a_b\)-learnable is \(\Sigma _4^0\)-complete and that the index set of TxtFex\(^*_*\)-learnable and the index set of TxtFext\(^*_*\)-learnable families are
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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$$AD_{\mathbb {R}}$$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.
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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
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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.
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Local reflection, definable elements and 1-provability Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-30 Evgeny Kolmakov
In this note we study several topics related to the schema of local reflection \(\textsf {Rfn} (T)\) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with \(\varSigma _n\)-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema
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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).
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Kurepa trees and spectra of $${\mathcal {L}}_{\omega _1,\omega }$$ L ω 1 , ω -sentences Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-29 Dima Sinapova, Ioannis Souldatos
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \({\mathcal {L}}_{\omega _1,\omega }\)-sentence \(\psi \) that codes Kurepa trees to prove the following statements: (1) The spectrum of \(\psi \) is consistently equal to \([\aleph _0,\aleph _{\omega _1}]\) and also consistently equal to \([\aleph _0,2^{\aleph _1})\), where \(2^{\aleph _1}\) is
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Bounded-low sets and the high/low hierarchy Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-27 Huishan Wu
Anderson and Csima (Notre Dame J Form Log 55:245–264, 2014) defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. (Arch Math Logic 56:507–521, 2017) constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable (c.e. for short). Ng and Yu (Notre Dame J Form Log, to appear) proved that bounded-high c.e. sets
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Classifying material implications over minimal logic Arch. Math. Logic (IF 0.485) Pub Date : 2020-03-07 Hannes Diener, Maarten McKubre-Jordens
The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent
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Chang’s Conjecture with $$\square _{\omega _1, 2}$$ □ ω 1 , 2 from an $$\omega _1$$ ω 1 -Erdős cardinal Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-21 Itay Neeman, John Susice
Answering a question of Sakai (Arch Math Logic 52(1–2):29–45, 2013), we show that the existence of an \(\omega _1\)-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with \(\square _{\omega _1, 2}\). By a result of Donder (In: Set theory and model theory (Bonn, 1979), volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give
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A version of $$\kappa $$ κ -Miller forcing Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-20 Heike Mildenberger, Saharon Shelah
We consider a version of \(\kappa \)-Miller forcing on an uncountable cardinal \(\kappa \). We show that under \(2^{<\kappa } = \kappa \) this forcing collapses \(2^\kappa \) to \(\omega \) and adds a \(\kappa \)-Cohen real. The same holds under the weaker assumptions that \({{\,\mathrm{cf}\,}}(\kappa ) > \omega \), \(2^{2^{<\kappa }}= 2^\kappa \), and forcing with \(([\kappa ]^\kappa , \subseteq )\)
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Tanaka’s theorem revisited Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-18 Saeideh Bahrami
Tanaka (Ann Pure Appl Log 84:41–49, 1997) proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \(({\mathcal {M}}, {\mathcal {A}})\) of the subsystem \(\mathrm {WKL}_{0}\) of second order arithmetic, and any element m of \({\mathcal {M}}\), there is a self-embedding j of \(({\mathcal {M}},{\mathcal {A}})\) onto a proper initial segment
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Classifying equivalence relations in the Ershov hierarchy Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-13 Nikolay Bazhenov, Manat Mustafa, Luca San Mauro, Andrea Sorbi, Mars Yamaleev
Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility \(\leqslant _c\). This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \(\Delta ^0_2\) case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable
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End extensions of models of fragments of PA Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-11 C. Dimitracopoulos, V. Paschalis
In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic (PA). In particular, we give alternative proofs that concern (a) a result of Clote (Fundam Math 127(2):163–170, 1986); (Fundam Math 158(3):301–302, 1998), on the end extendability of arbitrary models of \(\Sigma _n\)-induction, for \(n{\ge } 2\), and (b) the fact that
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On the independence of premiss axiom and rule Arch. Math. Logic (IF 0.485) Pub Date : 2020-02-07 Hajime Ishihara, Takako Nemoto
In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule (\(\mathrm {IPR}\)) for (many-sorted) intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \(\mathbf {HA}\) and \(\mathbf {HA}^\omega \) which are closed under \(\mathrm {IPR}\) but do
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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
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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
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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
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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
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Incomparability in local structures of s -degrees and Q -degrees Arch. Math. Logic (IF 0.485) Pub Date : 2020-01-23 Irakli Chitaia, Keng Meng Ng, Andrea Sorbi, Yue Yang
We show that for every intermediate \(\Sigma ^0_2\) s-degree (i.e. a nonzero s-degree strictly below the s-degree of the complement of the halting set) there exists an incomparable \(\Pi ^0_1\) s-degree. (The same proof yields a similar result for other positive reducibilities as well, including enumeration reducibility.) As a consequence, for every intermediate \(\Pi ^0_2\) Q-degree (i.e. a nonzero
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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
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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.
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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
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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.
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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
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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
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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
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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
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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 \(T^i_2\)
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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
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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.
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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.
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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
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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.
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