Year:

  1.  11
    Prikry Forcing and Tree Prikry Forcing of Various Filters.Tom Benhamou - 2019 - Archive for Mathematical Logic 58 (7-8):787-817.
    In this paper, we answer a question asked in Koepke et al. regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  2.  7
    Construction with Opposition: Cardinal Invariants and Games.Jörg Brendle, Michael Hrušák & Víctor Torres-Pérez - 2019 - Archive for Mathematical Logic 58 (7-8):943-963.
    We consider several game versions of the cardinal invariants \, \ and \. We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \ and \ are both relatively consistent with ZFC, where \ and \ are the principal game versions of \ and \, respectively. The corresponding question for \ remains open.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  3.  9
    Complete and Atomic Tarski Algebras.Sergio Arturo Celani - 2019 - Archive for Mathematical Logic 58 (7-8):899-914.
    Tarski algebras, also known as implication algebras or semi-boolean algebras, are the \-subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures \, where X is a non-empty set and \ is non-empty family of subsets of X such that \. This duality is a generalization of the known duality between sets (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  4.  8
    Free Sequences in $${Mathscr {P}}Left /Text {Fin}$$.David Chodounský, Vera Fischer & Jan Grebík - 2019 - Archive for Mathematical Logic 58 (7-8):1035-1051.
    We investigate maximal free sequences in the Boolean algebra \ {/}\text {fin}\), as defined by Monk :593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \. Answering a question of Monk, we demonstrate the consistency of \. In fact, this consistency is demonstrated in the model of Shelah for \ :433–443, 1992). Our paper provides a streamlined (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  5.  10
    Chainable and Circularly Chainable Semicomputable Sets in Computable Topological Spaces.Eugen Čičković, Zvonko Iljazović & Lucija Validžić - 2019 - Archive for Mathematical Logic 58 (7-8):885-897.
    We examine conditions under which, in a computable topological space, a semicomputable set is computable. It is known that in a computable metric space a semicomputable set S is computable if S is a continuum chainable from a to b, where a and b are computable points, or S is a circularly chainable continuum which is not chainable. We prove that this result holds in any computable topological space.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  6.  14
    Pseudo P-Points and Splitting Number.Alan Dow & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (7-8):1005-1027.
    We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  7.  13
    The Strong Tree Property and the Failure of SCH.Jin Du - 2019 - Archive for Mathematical Logic 58 (7-8):867-875.
    Fontanella :193–207, 2014) showed that if \ is an increasing sequence of supercompacts and \, then the strong tree property holds at \. Building on a proof by Neeman, we show that the strong tree property at \ is consistent with \, where \ is singular strong limit of countable cofinality.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  8.  7
    Definable Valuations Induced by Multiplicative Subgroups and NIP Fields.Katharina Dupont, Assaf Hasson & Salma Kuhlmann - 2019 - Archive for Mathematical Logic 58 (7-8):819-839.
    We study the algebraic implications of the non-independence property and variants thereof on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” :1850007, 2018).
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  9.  12
    The Spectrum of Independence.Vera Fischer & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (7-8):877-884.
    We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote \\). Here mif abbreviates maximal independent family. We show that:1.whenever \ are finitely many regular uncountable cardinals, it is consistent that \\); 2.whenever \ has uncountable cofinality, it is consistent that \=\{\aleph _1,\kappa =\mathfrak {c}\}\). Assuming large cardinals, in addition to above, we can provide that $$\begin{aligned} \cap \hbox {Spec}=\emptyset \end{aligned}$$for each i, \.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  10.  8
    Set-Theoretic Blockchains.Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner & Kameryn J. Williams - 2019 - Archive for Mathematical Logic 58 (7-8):965-997.
    Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  11.  13
    Using Ramsey’s Theorem Once.Jeffry L. Hirst & Carl Mummert - 2019 - Archive for Mathematical Logic 58 (7-8):857-866.
    We show that \\) cannot be proved with one typical application of \\) in an intuitionistic extension of \ to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  12.  11
    A Note on Groups Definable in the P -Adic Field.Anand Pillay & Ningyuan Yao - 2019 - Archive for Mathematical Logic 58 (7-8):1029-1034.
    It is known Hrushovski and Pillay that a group G definable in the field \ of p-adic numbers is definably locally isomorphic to the group \\) of p-adic points of a algebraic group H over \. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in \ is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general context (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  13.  8
    An Inner Model Theoretic Proof of Becker’s Theorem.Grigor Sargsyan - 2019 - Archive for Mathematical Logic 58 (7-8):999-1003.
    We re-prove Becker’s theorem from Becker :229–234, 1981) by showing that \}\) implies that \\vDash ``\omega _2\) is Open image in new window -supercompact”. Our proof uses inner model theoretic tools instead of Baire category. We also show that \ is \-strongly compact.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  14.  12
    Generic Vopěnka Cardinals and Models of ZF with Few $$Aleph _1$$-Suslin Sets.Trevor M. Wilson - 2019 - Archive for Mathematical Logic 58 (7-8):841-856.
    We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  15.  8
    Continuous Triangular Norm Based Fuzzy Topology.Dexue Zhang & Gao Zhang - 2019 - Archive for Mathematical Logic 58 (7-8):915-942.
    For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives rise to an autonomous (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  16.  17
    Degrees of Bi-Embeddable Categoricity of Equivalence Structures.Nikolay Bazhenov, Ekaterina Fokina, Dino Rossegger & Luca San Mauro - 2019 - Archive for Mathematical Logic 58 (5-6):543-563.
    We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, \ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \ bi-embeddable categoricity and relative \ bi-embeddable categoricity coincide for equivalence structures for \. We also prove that computable equivalence structures have degree of bi-embeddable categoricity \, or \. We furthermore obtain results (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  17.  10
    VC-Density for Trees.Anton Bobkov - 2019 - Archive for Mathematical Logic 58 (5-6):587-603.
    We show that in the theory of infinite trees the VC-function is optimal. This generalizes a result of Simon showing that trees are dp-minimal.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  18.  10
    On the Classification of Vertex-Transitive Structures.John Clemens, Samuel Coskey & Stephanie Potter - 2019 - Archive for Mathematical Logic 58 (5-6):565-574.
    We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above \ in complexity.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  19.  10
    Reverse Mathematics and Colorings of Hypergraphs.Caleb Davis, Jeffry Hirst, Jake Pardo & Tim Ransom - 2019 - Archive for Mathematical Logic 58 (5-6):575-585.
    Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to \, \, and \.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  20.  12
    Truth, Disjunction, and Induction.Ali Enayat & Fedor Pakhomov - 2019 - Archive for Mathematical Logic 58 (5-6):753-766.
    By a well-known result of Kotlarski et al., first-order Peano arithmetic \ can be conservatively extended to the theory \ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \ while maintaining conservativity over \. Our main result shows that conservativity fails even (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  21.  10
    Ideals of Independence.Vera Fischer & Diana Carolina Montoya - 2019 - Archive for Mathematical Logic 58 (5-6):767-785.
    We study two ideals which are naturally associated to independent families. The first of them, denoted \, is characterized by a diagonalization property which allows along a cofinal sequence of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted \\), originates in Shelah’s proof of \ in Shelah, 433–443, 1992). We show that for every independent family \, \\subseteq \mathcal {J}_\mathcal {A}\) and define a class of maximal independent families, to which we refer (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  22.  12
    Non-Homogeneity of Quotients of Prikry Forcings.Moti Gitik & Eyal Kaplan - 2019 - Archive for Mathematical Logic 58 (5-6):649-710.
    We study non-homogeneity of quotients of Prikry and tree Prikry forcings with non-normal ultrafilters over some natural distributive forcing notions.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  23.  8
    Determinacy Separations for Class Games.Sherwood Hachtman - 2019 - Archive for Mathematical Logic 58 (5-6):635-648.
    We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  24.  11
    A Flexible Type System for the Small Veblen Ordinal.Florian Ranzi & Thomas Strahm - 2019 - Archive for Mathematical Logic 58 (5-6):711-751.
    We introduce and analyze two theories for typed inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal \. We investigate on the one hand the applicative theory \ of functions, inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system \\). On the other hand, we investigate the arithmetical theory \ of typed inductive definitions, a natural subsystem of \, and carry out a wellordering proof within \ (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  25.  17
    Cut Elimination for Entailment Relations.Davide Rinaldi & Daniel Wessel - 2019 - Archive for Mathematical Logic 58 (5-6):605-625.
    Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  26.  10
    Comparing the Degrees of Enumerability and the Closed Medvedev Degrees.Paul Shafer & Andrea Sorbi - 2019 - Archive for Mathematical Logic 58 (5-6):527-542.
    We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  27.  11
    A Kuroda-Style J -Translation.Benno van den Berg - 2019 - Archive for Mathematical Logic 58 (5-6):627-634.
    A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  28.  8
    Elementary Theories and Hereditary Undecidability for Semilattices of Numberings.Nikolay Bazhenov, Manat Mustafa & Mars Yamaleev - 2019 - Archive for Mathematical Logic 58 (3-4):485-500.
    A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their first order theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. In this paper, we study decidability for theories of upper semilattices that arise from the theory of numberings. We use the following approach: given a level of complexity, say \, we consider the upper semilattice \ of (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  29.  9
    Well Quasi Orders in a Categorical Setting.Marco Benini & Roberta Bonacina - 2019 - Archive for Mathematical Logic 58 (3-4):501-526.
    This article describes well quasi orders as a category, focusing on limits and colimits. In particular, while quasi orders with monotone maps form a category which is finitely complete, finitely cocomplete, and with exponentiation, the full subcategory of well quasi orders is finitely complete and cocomplete, but with no exponentiation. It is interesting to notice how finite antichains and finite proper descending chains interact to induce this structure in the category: in fact, the full subcategory of quasi orders with finite (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  30.  9
    Conjugacy for Homogeneous Ordered Graphs.Samuel Coskey & Paul Ellis - 2019 - Archive for Mathematical Logic 58 (3-4):457-467.
    We show that for any countable homogeneous ordered graph G, the conjugacy problem for automorphisms of G is Borel complete. In fact we establish that each such G satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of G is Borel reducible to the conjugacy relation on automorphisms of G.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  31.  10
    On Uniformly Continuous Functions Between Pseudometric Spaces and the Axiom of Countable Choice.Samuel G. da Silva - 2019 - Archive for Mathematical Logic 58 (3-4):353-358.
    In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  32.  8
    Some Remarks on Inp-Minimal and Finite Burden Groups.Jan Dobrowolski & John Goodrick - 2019 - Archive for Mathematical Logic 58 (3-4):267-274.
    We prove that any left-ordered inp-minimal group is abelian and we provide an example of a non-abelian left-ordered group of dp-rank 2. Furthermore, we establish a necessary condition for a group to have finite burden involving normalizers of definable sets, reminiscent of other chain conditions for stable groups.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  33.  9
    Diamond, Scales and GCH Down to $$\Aleph _{\Omega ^2}$$ ℵ Ω 2.Jin Du - 2019 - Archive for Mathematical Logic 58 (3-4):427-442.
    Gitik and Rinot :1771–1795, 2012) proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal \ of countable cofinality such that \, there is a very good scale at \, and \ fails along some reflecting stationary subset of \\). In this paper, we force over Gitik and Rinot’s model but with a modification of Gitik–Sharon :311, 2008) diagonal Prikry forcing to get this result for \.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  34.  6
    A Note on Iterated Consistency and Infinite Proofs.Anton Freund - 2019 - Archive for Mathematical Logic 58 (3-4):339-346.
    Schmerl and Beklemishev’s work on iterated reflection achieves two aims: it introduces the important notion of \-ordinal, characterizing the \-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the \-ordinals for a range of theories. The present note demonstrates that these achievements are independent: we read off \-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  35.  9
    Distinct Volume Subsets Via Indiscernibles.William Gasarch & Douglas Ulrich - 2019 - Archive for Mathematical Logic 58 (3-4):469-483.
    Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  36.  5
    On Positive Local Combinatorial Dividing-Lines in Model Theory.Vincent Guingona & Cameron Donnay Hill - 2019 - Archive for Mathematical Logic 58 (3-4):289-323.
    We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  37.  9
    A Laver-Like Indestructibility for Hypermeasurable Cardinals.Radek Honzik - 2019 - Archive for Mathematical Logic 58 (3-4):275-287.
    We show that if \ is \\)-hypermeasurable for some cardinal \ with \ \le \mu \) and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model \ in which the \\)-hypermeasurability of \ is indestructible by the Cohen forcing at \ of any length up to \ is \\)-hypermeasurable in \). The preservation of hypermeasurability is useful for subsequent arguments. The construction of \ is based on the ideas of Woodin and Cummings :1–39, (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  38.  7
    On the Non-Existence of Mad Families.Haim Horowitz & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (3-4):325-338.
    We show that the non-existence of mad families is equiconsistent with \, answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that \ “there is no maximal independent set in G” is equiconsistent with \ “there exists an inaccessible cardinal”.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  39.  10
    A Note on Gentzen’s Ordinal Assignment.Annika Kanckos - 2019 - Archive for Mathematical Logic 58 (3-4):347-352.
    Gentzen’s height measure of the 1938 consistency proof is a cumulative complexity measure for sequents that is measured bottom-up in a derivation. By a factorisation of the ordinal assignment a top-down ordinal assignment can be given that does not depend on information occurring below the sequent to which the ordinal is assigned. Furthermore, an ordinal collapsing function is defined in order to collapse the top-down ordinal to the one assigned by Gentzen’s own ordinal assignment. A direct definition of the factorised (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  40.  7
    A Continuity Principle Equivalent to the Monotone $$Pi ^{0}_{1}$$ Fan Theorem.Tatsuji Kawai - 2019 - Archive for Mathematical Logic 58 (3-4):443-456.
    The strong continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone \ bars. We work in the context of constructive reverse mathematics.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  41.  10
    Maehara-Style Modal Nested Calculi.Roman Kuznets & Lutz Straßburger - 2019 - Archive for Mathematical Logic 58 (3-4):359-385.
    We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness (...)
    No categories
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  42.  5
    Degree Spectra of Real Closed Fields.Russell Miller & Victor Ocasio González - 2019 - Archive for Mathematical Logic 58 (3-4):387-411.
    Several researchers have recently established that for every Turing degree \, the real closed field of all \-computable real numbers has spectrum \. We investigate the spectra of real closed fields further, focusing first on subfields of the field \ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of \ with no computable copy. Then we (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  43.  12
    Dickson’s Lemma and Weak Ramsey Theory.Yasuhiko Omata & Florian Pelupessy - 2019 - Archive for Mathematical Logic 58 (3-4):413-425.
    We explore the connections between Dickson’s lemma and weak Ramsey theory. We show that a weak version of the Paris–Harrington principle for pairs in c colors and miniaturized Dickson’s lemma for c-tuples are equivalent over \. Furthermore, we look at a cascade of consequences for several variants of weak Ramsey’s theorem.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  44.  6
    An Extension of Shelah’s Trichotomy Theorem.Shehzad Ahmed - 2019 - Archive for Mathematical Logic 58 (1-2):137-153.
    Shelah develops the theory of \\) without the assumption that \\), going so far as to get generators for every \\) under some assumptions on I. Our main theorem is that we can also generalize Shelah’s trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for \\) which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  45.  12
    The Iterability Hierarchy Above $${{\Mathrm{\Mathsf {I3}}}}$$ I 3.Alessandro Andretta & Vincenzo Dimonte - 2019 - Archive for Mathematical Logic 58 (1-2):77-97.
    In this paper we introduce a new hierarchy of large cardinals between \ and \, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  46.  9
    A Completeness Theorem for Continuous Predicate Modal Logic.Stefano Baratella - 2019 - Archive for Mathematical Logic 58 (1-2):183-201.
    We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen :168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen, that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  47.  12
    The Binary Expansion and the Intermediate Value Theorem in Constructive Reverse Mathematics.Josef Berger, Hajime Ishihara, Takayuki Kihara & Takako Nemoto - 2019 - Archive for Mathematical Logic 58 (1-2):203-217.
    We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  48.  5
    Convexity and Unique Minimum Points.Josef Berger & Gregor Svindland - 2019 - Archive for Mathematical Logic 58 (1-2):27-34.
    We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  49.  10
    Sequent Calculus for Classical Logic Probabilized.Marija Boričić - 2019 - Archive for Mathematical Logic 58 (1-2):119-136.
    Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \ with the intended meaning that “the probability of truthfulness of \ belongs to the interval [a, b]”. This method makes it possible to define a system of derivations (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  50.  11
    Diagonal Reflections on Squares.Gunter Fuchs - 2019 - Archive for Mathematical Logic 58 (1-2):1-26.
    The effects of the forcing axioms \, \ and \ on the failure of weak threaded square principles of the form \\) are analyzed. To this end, a diagonal reflection principle, \, and it implies the failure of \\) if \. It is also shown that this result is sharp. It is noted that \/\ imply the failure of \\), for every regular \, and that this result is sharp as well.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  51.  9
    Strange Ultrafilters.Moti Gitik - 2019 - Archive for Mathematical Logic 58 (1-2):35-52.
    We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  52.  11
    A Model of the Generic Vopěnka Principle in Which the Ordinals Are Not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  53.  14
    Selfextensional Logics with a Distributive Nearlattice Term.Luciano J. González - 2019 - Archive for Mathematical Logic 58 (1-2):219-243.
    We define when a ternary term m of an algebraic language \ is called a distributive nearlattice term -term) of a sentential logic \. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras associated with a selfextensional logic with a \-term is a variety, and we obtain (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  54.  9
    Uniform Interpolation and Sequent Calculi in Modal Logic.Rosalie Iemhoff - 2019 - Archive for Mathematical Logic 58 (1-2):155-181.
    A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent calculi cannot exist.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  55.  8
    A Strong Failure of $$\Aleph _0$$ ℵ 0 -Stability for Atomic Classes.Michael C. Laskowski & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (1-2):99-118.
    We study classes of atomic models \ of a countable, complete first-order theory T. We prove that if \ is not \-small, i.e., there is an atomic model N that realizes uncountably many types over \\) for some finite \ from N, then there are \ non-isomorphic atomic models of T, each of size \.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  56.  10
    Families of Sets Related to Rosenthal’s Lemma.Damian Sobota - 2019 - Archive for Mathematical Logic 58 (1-2):53-69.
    A family \ is called Rosenthal if for every Boolean algebra \, bounded sequence \ of measures on \, antichain \ in \, and \, there exists \ such that \<\varepsilon \) for every \. Well-known and important Rosenthal’s lemma states that \ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  57.  7
    Extendible Cardinals and the Mantle.Toshimichi Usuba - 2019 - Archive for Mathematical Logic 58 (1-2):71-75.
    The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
 Previous issues
  
Next issues