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  1.  8
    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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  2.  7
    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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  3.  19
    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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  4.  1
    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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  5.  4
    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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  6.  3
    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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  7.  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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  8.  2
    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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  9.  4
    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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  10.  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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  11.  7
    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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  12.  1
    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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  13.  8
    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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  14.  2
    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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  15.  10
    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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  16.  11
    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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  17.  12
    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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  18.  7
    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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  19.  13
    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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  20.  14
    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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  21.  12
    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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  22.  11
    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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  23.  9
    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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  24.  9
    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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  25.  13
    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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