Year:

  1.  4
    Tanaka’s theorem revisited.Saeideh Bahrami - 2020 - Archive for Mathematical Logic 59 (7):865-877.
    Tanaka proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \\) of the subsystem \ of second order arithmetic, and any element m of \, there is a self-embedding j of \\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \\ \)of \ and a proper cut \ of \:Theorem A. The (...)
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  2.  7
    Classifying equivalence relations in the Ershov hierarchy.Nikolay Bazhenov, Manat Mustafa, Luca San Mauro, Andrea Sorbi & Mars Yamaleev - 2020 - Archive for Mathematical Logic 59 (7):835-864.
    Computably enumerable equivalence relations received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility \. This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \ case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by \ on the \ equivalence relations. (...)
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  3.  2
    Fields with automorphism and valuation.Özlem Beyarslan, Daniel Max Hoffmann, Gönenç Onay & David Pierce - 2020 - Archive for Mathematical Logic 59 (7):997-1008.
    The model companion of the theory of fields with valuation and automorphism exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion.
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  4.  3
    Incomparability in local structures of s -degrees and Q -degrees.Irakli Chitaia, Keng Meng Ng, Andrea Sorbi & Yue Yang - 2020 - Archive for Mathematical Logic 59 (7):777-791.
    We show that for every intermediate \ s-degree there exists an incomparable \ s-degree. As a consequence, for every intermediate \ Q-degree there exists an incomparable \ Q-degree. We also show how these results can be applied to provide proofs or new proofs of upper density results in local structures of s-degrees and Q-degrees.
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  5.  8
    Classifying material implications over minimal logic.Hannes Diener & Maarten McKubre-Jordens - 2020 - Archive for Mathematical Logic 59 (7):905-924.
    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 groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...)
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  6.  3
    End extensions of models of fragments of PA.C. Dimitracopoulos & V. Paschalis - 2020 - Archive for Mathematical Logic 59 (7):817-833.
    In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic. In particular, we give alternative proofs that concern a result of Clote :163–170, 1986); :301–302, 1998), on the end extendability of arbitrary models of \-induction, for \, and the fact that every model of \-induction has a proper end extension satisfying \-induction; although this fact was not explicitly stated before, it follows by earlier results of Enayat and Wong and (...)
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  7.  8
    On the independence of premiss axiom and rule.Hajime Ishihara & Takako Nemoto - 2020 - Archive for Mathematical Logic 59 (7):793-815.
    In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule ) for intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \ and \ which are closed under \ but do not derive the independence of premiss axiom.
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  8.  1
    Local Reflection, Definable Elements and 1-Provability.Evgeny Kolmakov - 2020 - Archive for Mathematical Logic 59 (7-8):979-996.
    In this note we study several topics related to the schema of local reflection \\) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with \-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema we give a new model-theoretic proof of the \-conservativity of uniform \-reflection over relativized local \-reflection. We also study the proof-theoretic strength of Feferman’s theorem, i.e., the assertion of 1-provability (...)
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  9.  4
    A version of $$kappa $$ κ -Miller forcing.Heike Mildenberger & Saharon Shelah - 2020 - Archive for Mathematical Logic 59 (7):879-892.
    We consider a version of \-Miller forcing on an uncountable cardinal \. We show that under \, \, and forcing with \\) collapses \ to \.
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  10.  6
    Chang’s Conjecture with $$square {omega _1, 2}$$ □ ω 1, 2 from an $$omega 1$$ ω 1 -Erdős cardinal.Itay Neeman & John Susice - 2020 - Archive for Mathematical Logic 59 (7):893-904.
    Answering a question of Sakai :29–45, 2013), we show that the existence of an \-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with \. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of \ and \ \twoheadrightarrow \) for uncountable \.
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  11.  4
    Chang’s Conjecture with $$Square {Omega _1, 2}$$ □ Ω 1, 2 From an $$Omega 1$$ Ω 1 -Erdős Cardinal.Itay Neeman & John Susice - 2020 - Archive for Mathematical Logic 59 (7-8):893-904.
    Answering a question of Sakai :29–45, 2013), we show that the existence of an ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with □ω1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\omega _1, 2}$$\end{document}. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to (...)
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  12.  4
    Special ultrafilters and cofinal subsets of $$({}^omega omega, <^*)$$.Peter Nyikos - 2020 - Archive for Mathematical Logic 59 (7):1009-1026.
    The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...)
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  13.  1
    Ideal Generalizations of Egoroff’s Theorem.Miroslav Repický - 2020 - Archive for Mathematical Logic 59 (7-8):957-977.
    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 :269–282, 2017).
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  14.  2
    Easton collapses and a strongly saturated filter.Masahiro Shioya - 2020 - Archive for Mathematical Logic 59 (7):1027-1036.
    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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  15.  3
    Kurepa trees and spectra of $${mathcal {L}}{omega 1,omega }$$ L ω 1, ω -sentences.Dima Sinapova & Ioannis Souldatos - 2020 - Archive for Mathematical Logic 59 (7):939-956.
    We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \-sentence \ that codes Kurepa trees to prove the following statements: The spectrum of \ is consistently equal to \ and also consistently equal to \\), where \ is weakly inaccessible.The amalgamation spectrum of \ is consistently equal to \ and \\), where again \ is weakly inaccessible. This is the first example of an \-sentence whose spectrum and amalgamation spectrum are consistently both right-open and (...)
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  16.  5
    Bounded-Low Sets and the High/Low Hierarchy.Huishan Wu - 2020 - Archive for Mathematical Logic 59 (7-8):925-938.
    Anderson and Csima defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable. Ng and Yu proved that bounded-high c.e. sets are Turing complete, thus answered the conjecture positively. Wu and Wu showed that bounded-low sets can be superhigh by constructing a Turing complete bounded-low c.e. set. In this paper, we continue the study of the comparison between the bounded-jump and (...)
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  17.  3
    Scattered sentences have few separable randomizations.Uri Andrews, Isaac Goldbring, Sherwood Hachtman, H. Jerome Keisler & David Marker - 2020 - Archive for Mathematical Logic 59 (5):743-754.
    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 \-sentence with few separable randomizations has countably many countable models.
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  18.  3
    A diamond-plus principle consistent with AD.Daniel W. Cunningham - 2020 - Archive for Mathematical Logic 59 (5):755-775.
    After showing that \ refutes \ for all regular cardinals \, we present a diamond-plus principle \ concerning all subsets of \. Using a forcing argument, we prove that \ holds in Steel’s core model \}}\), an inner model in which the axiom of determinacy can hold. The combinatorial principle \ is then extended, in \}}\), to successor cardinals \ and to certain cardinals \ that are not ineffable. Here \ is the supremum of the ordinals that are the surjective (...)
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  19.  6
    Completeness of the primitive recursive $$omega $$ ω -rule.Emanuele Frittaion - 2020 - Archive for Mathematical Logic 59 (5):715-731.
    Shoenfield’s completeness theorem states that every true first order arithmetical sentence has a recursive \-proof encodable by using recursive applications of the \-rule. For a suitable encoding of Gentzen style \-proofs, we show that Shoenfield’s completeness theorem applies to cut free \-proofs encodable by using primitive recursive applications of the \-rule. We also show that the set of codes of \-proofs, whether it is based on recursive or primitive recursive applications of the \-rule, is \ complete. The same \ completeness (...)
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  20.  8
    Rank-initial embeddings of non-standard models of set theory.Paul Kindvall Gorbow - 2020 - Archive for Mathematical Logic 59 (5):517-563.
    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 carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment (...)
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  21.  5
    On Ramsey choice and partial choice for infinite families of n -element sets.Lorenz Halbeisen & Eleftherios Tachtsis - 2020 - Archive for Mathematical Logic 59 (5):583-606.
    For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ 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 \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...)
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  22.  3
    Uniform Lyndon interpolation property in propositional modal logics.Taishi Kurahashi - 2020 - Archive for Mathematical Logic 59 (5):659-678.
    We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of uniform interpolants for \ (...)
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  23.  6
    Reversibility of extreme relational structures.Miloš S. Kurilić & Nenad Morača - 2020 - Archive for Mathematical Logic 59 (5):565-582.
    A relational structure \ is called reversible iff each bijective homomorphism from \ onto \ 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 of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable \-theory defines a class of interpretations (...)
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  24.  6
    Finite sets and infinite sets in weak intuitionistic arithmetic.Takako Nemoto - 2020 - Archive for Mathematical Logic 59 (5):607-657.
    In this paper, we consider, for a set \ of natural numbers, the following notions of finitenessFIN1:There are a natural number l and a bijection f between \\);FIN5:It is not the case that \\), and infinitenessINF1:There are not a natural number l and a bijection f between \\);INF5:\\). In this paper, we systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, (...)
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  25.  4
    Some nondefinability results with entire functions in a polynomially bounded o-minimal structure.Hassan Sfouli - 2020 - Archive for Mathematical Logic 59 (5):733-741.
    Let \=\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 \ is not definable in the ordered field of real numbers with restricted analytic functions, \. Furthermore, we show that there is \ such that the function \\) on \ is not definable in \, where \ the expansion of the real field generated by multisummable real series.
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  26.  3
    Definable combinatorics with dense linear orders.Himanshu Shukla, Arihant Jain & Amit Kuber - 2020 - Archive for Mathematical Logic 59 (5):679-701.
    We compute the model-theoretic Grothendieck ring, \\), of a dense linear order with or without end points, \\), as a structure of the signature \, and show that it is a quotient of the polynomial ring over \ generated by \\) by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over \ generated by \. As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable (...)
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  27.  2
    The noneffectivity of Arslanov’s completeness criterion and related theorems.Sebastiaan A. Terwijn - 2020 - Archive for Mathematical Logic 59 (5):703-713.
    We discuss the 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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  28.  9
    Proof-Theoretic Strengths of the Well-Ordering Principles.Toshiyasu Arai - 2020 - Archive for Mathematical Logic 59 (3-4):257-275.
    In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions \ on ordinals is shown to be equal to the least fixed point of \. Moreover corrections to the previous paper are made.
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  29.  10
    Covering properties of $$omega $$ω -mad families.Leandro Aurichi & Lyubomyr Zdomskyy - 2020 - Archive for Mathematical Logic 59 (3):445-452.
    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 \ 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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  30.  25
    Weaker variants of infinite time Turing machines.Matteo Bianchetti - 2020 - Archive for Mathematical Logic 59 (3):335-365.
    Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the lim sup of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the lim sup rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content (...)
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  31.  3
    Detecting properties from descriptions of groups.Iva Bilanovic, Jennifer Chubb & Sam Roven - 2020 - Archive for Mathematical Logic 59 (3):293-312.
    We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense. We consider descriptions of two sorts: first, recursive presentations in terms of (...)
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  32.  6
    Square below a non-weakly compact cardinal.Hazel Brickhill - 2020 - Archive for Mathematical Logic 59 (3):409-426.
    In his seminal paper introducing the fine structure of L, Jensen proved that under \ 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 natural and simpler alternative to fine structure theory introduced by Friedman and Koepke.
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  33.  4
    Analytic Computable Structure Theory and $$L^P$$Lp -Spaces Part 2.Tyler Brown & Timothy H. McNicholl - 2020 - Archive for Mathematical Logic 59 (3-4):427-443.
    Suppose \ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \ 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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  34.  3
    Antichains of perfect and splitting trees.Paul Hein & Otmar Spinas - 2020 - Archive for Mathematical Logic 59 (3):367-388.
    We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.
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  35.  3
    Induction rules in bounded arithmetic.Emil Jeřábek - 2020 - Archive for Mathematical Logic 59 (3):461-501.
    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 \ induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems and \ of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.
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  36.  6
    Absorbing the structural rules in the sequent calculus with additional atomic rules.Franco Parlamento & Flavio Previale - 2020 - Archive for Mathematical Logic 59 (3):389-408.
    We show that if the structural rules are admissible over a set \ of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in \ to the multisuccedent minimal and intuitionistic \ 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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  37.  5
    Dependent choice as a termination principle.Thomas Powell - 2020 - Archive for Mathematical Logic 59 (3):503-516.
    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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  38.  10
    Scott sentences for equivalence structures.Sara B. Quinn - 2020 - Archive for Mathematical Logic 59 (3):453-460.
    For a computable structure \, 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 \\). If we can also show that \\) 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 for the structure. There are results that suggest that these complexities will always match. However, it was shown in (...)
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  39.  2
    A small ultrafilter number at smaller cardinals.Dilip Raghavan & Saharon Shelah - 2020 - Archive for Mathematical Logic 59 (3):325-334.
    It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on \ which is generated by fewer than \ sets.
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  40.  10
    Ordinal analyses for monotone and cofinal transfinite inductions.Kentaro Sato - 2020 - Archive for Mathematical Logic 59 (3):277-291.
    We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \ formulae are provable in \. It is shown to be larger than the proof-theoretic ordinal \ by power of base 2. (...)
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  41.  3
    On the Forking Topology of a Reduct of a Simple Theory.Ziv Shami - 2020 - Archive for Mathematical Logic 59 (3-4):313-324.
    Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In (...)
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  42.  11
    Properties of the atoms in finitely supported structures.Andrei Alexandru & Gabriel Ciobanu - 2020 - Archive for Mathematical Logic 59 (1):229-256.
    The goal of this paper is to present a collection of properties of the set of atoms and the set of finite injective tuples of atoms, as well as of the powersets of atoms in the framework of finitely supported structures. Some properties of atoms are obtained by translating classical Zermelo–Fraenkel results into the new framework, but several important properties are specific to finitely supported structures.
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  43.  12
    Fields with a dense-codense linearly independent multiplicative subgroup.Alexander Berenstein & Evgueni Vassiliev - 2020 - Archive for Mathematical Logic 59 (1):197-228.
    We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...)
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  44.  13
    Polynomial time ultrapowers and the consistency of circuit lower bounds.Jan Bydžovský & Moritz Müller - 2020 - Archive for Mathematical Logic 59 (1):127-147.
    A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory \ of all polynomial time functions. Generalizing a theorem of Hirschfeld :111–126, 1975), we show that every countable model of \ is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler Ultrafilters across (...)
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  45.  8
    The weakly compact reflection principle need not imply a high order of weak compactness.Brent Cody & Hiroshi Sakai - 2020 - Archive for Mathematical Logic 59 (1):179-196.
    The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \ then there is a forcing extension preserving (...)
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  46.  14
    Compactness in MV-topologies: Tychonoff theorem and Stone–Čech compactification.Luz Victoria De La Pava & Ciro Russo - 2020 - Archive for Mathematical Logic 59 (1):57-79.
    In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and Lowen’s (...)
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  47.  15
    NIP Henselian Valued Fields.Franziska Jahnke & Pierre Simon - 2020 - Archive for Mathematical Logic 59 (1-2):167-178.
    We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if is a henselian valued field of residue characteristic \=p\) such that if \, depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then is NIP iff Kv is NIP and v is roughly separably (...)
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  48.  13
    Non-Forking W-Good Frames.Marcos Mazari-Armida - 2020 - Archive for Mathematical Logic 59 (1-2):31-56.
    We introduce the notion of a w-good \-frame which is a weakening of Shelah’s notion of a good \-frame. Existence of a w-good \-frame implies existence of a model of size \. Tameness and amalgamation imply extension of a w-good \-frame to larger models. As an application we show:Theorem 0.1. Suppose\. If \ = \mathbb {I} = 1 \le \mathbb {I} < 2^{\lambda ^{++}}\)and\is\\)-tame, then\.The proof presented clarifies some of the details of the main theorem of Shelah and avoids using (...)
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  49.  13
    Product of Invariant Types Modulo Domination–Equivalence.Rosario Mennuni - 2020 - Archive for Mathematical Logic 59 (1-2):1-29.
    We investigate the interaction between the product of invariant types and domination–equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.
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  50.  10
    The subcompleteness of diagonal Prikry forcing.Kaethe Minden - 2020 - Archive for Mathematical Logic 59 (1):81-102.
    Let \ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in \ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in \, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to \ is subcomplete above \, where \ is any regular cardinal below the first (...)
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  51.  10
    Definable One-Dimensional Topologies in O-Minimal Structures.Ya’Acov Peterzil & Ayala Rosel - 2020 - Archive for Mathematical Logic 59 (1-2):103-125.
    We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space \ \) is definably homeomorphic to an affine definable space with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.
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  52.  14
    Deciding Active Structural Completeness.Michał M. Stronkowski - 2020 - Archive for Mathematical Logic 59 (1-2):149-165.
    We prove that if an n-element algebra generates the variety \ which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in \ is at most \\cdot n^{2\cdot n}}\). As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra \ generates the structurally complete variety \\) in the cases when \\) is congruence modular or \\) is congruence meet-semidistributive or \ is a semigroup.
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