Showing papers in "Mathematical Logic Quarterly in 2011"
TL;DR: It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails, and the coincidence between triviality and lowness notions for truth-table Schnorrrandomness is established.
Abstract: Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined in 6 too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
25 citations
TL;DR: The notion of the radical of a filter in BL-algebras is defined and several characterizations of theradical are given and it is proved that A/F is an MV-algebra if and only if Ds(A) ⊆ F is a semi maximal filter of A.
Abstract: In this paper, the notion of the radical of a filter in BL-algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV-algebra if and only if Ds(A) ⊆ F. After that we define the notion of semi maximal filter in BL-algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL-algebra. Moreover, we prove that A/F is a semi simple BL-algebra if and only if F is a semi maximal filter of A. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
19 citations
TL;DR: A general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space.
Abstract: We present a general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to be useful as a combinatorial tool for the study of general reducibilities for subsets of the Baire space [10] (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
16 citations
TL;DR: In this paper, notions of fuzzy closure system and fuzzy closure L—system on L—ordered sets are introduced from the fuzzy point of view and the correspondence between fuzzy closure systems and fuzzy Galois operators is established.
Abstract: In this paper, notions of fuzzy closure system and fuzzy closure L—system on L—ordered sets are introduced from the fuzzy point of view. We first explore the fundamental properties of fuzzy closure systems. Then the correspondence between fuzzy closure systems (fuzzy closure L—systems) and fuzzy closure operators is established. Finally, we study the connections between fuzzy closure systems and fuzzy Galois connections. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
16 citations
TL;DR: This paper develops an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MVs devised by Filipoiu and Georgescu and by Dubuc and Poveda.
Abstract: In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein ([2]), we develop an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MV-algebras devised by Filipoiu and Georgescu ([18]) and by Dubuc and Poveda ([16]). Following Davey approach ([12]), we use a subdirect representation of MV-algebras that is based on local MV-algebras. This allowed us to obtain:
(a) a representation of any MV-algebras as MV-algebra of all global sections of a sheaf of local MV-algebras on the spectruum of its prime ideals;
(b) a representation of MV-algebras, having the space of minimal prime ideals compact, as MV-algebra of all global sections of a Hausdorff sheaf of MV-chains on the space of minimal prime ideals, which is a Stone space;
(c) an adjunction between the category of all MV-algebras and the category of MV-algebraic spaces, where an MV-algebraic space is a pair (X, F), where X is a compact topological space and F is a sheaf of MV-algebras with stalks that are local (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
14 citations
TL;DR: It is shown that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no (proper, fine, κ-complete) ideal on Pκ(λ) is weakly λ+-saturated.
Abstract: We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no (proper, fine, κ-complete) ideal on Pκ(λ) is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
13 citations
TL;DR: In this article, it was shown that BWweak is instance-wise equivalent to the weak Konig's lemma for Σ01-trees (Σ 01- WKL) and that it does not solve the halting problem and does not lead to more than primitive recursive growth.
Abstract: The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstras principle (BW) and a weak variant of it.
We show that BW is instance-wise equivalent to the weak Konig’s lemma for Σ01-trees (Σ01- WKL). This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing an accumulation point for all bounded computable sequences.
Let BWweak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BWweak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and—using this—obtain a classification of the computational and logical strength of BWweak. Especially we show that BWweak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BWweak. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
13 citations
TL;DR: A number of natural correspondences are exhibited between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces and Tameness of Galois types emerges as a topological separation principle.
Abstract: We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
12 citations
TL;DR: Using games, as introduced by Hirsch and Hodkinson in algebraic logic, a recursive axiomatization of the class RQPEAα of representable quasi-polyadic equality algebras of any dimension α is given.
Abstract: Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class RQPEAα of representable quasi-polyadic equality algebras of any dimension α Following Sain and Thompson in modifying Andreka’s methods of splitting, to adapt the quasi-polyadic equality case, we show that if Σ is a set of equations axiomatizing RPEAn for and , k′ < ω are natural numbers, then Σ contains infinitely equations in which − occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur © 2011 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim
12 citations
TL;DR: A concept of ‘Horn sentence’ is defined in MVL and based upon the study of quasi-varieties of MVL models, the existence of initial models for MVL “Horn theories” is derived.
Abstract: We extend the concept of quasi-variety of first-order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi-varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi-varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
11 citations
TL;DR: This paper provides new characterizations for Q1 and Q2 that apply to almost all promise classes and languages L, thus creating a unifying framework for the study of these practically relevant questions.
Abstract: In this paper we investigate the following two questions:
Q1: Do there exist optimal proof systems for a given language L?
Q2: Do there exist complete problems for a given promise class ?
For concrete languages L (such as TAUT or SAT) and concrete promise classes (such as , , , disjoint -pairs etc.), these questions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new characterizations for Q1 and Q2 that apply to almost all promise classes and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount of advice is available in the underlying machine model. For promise classes with promise condition in , the advice can be replaced by a tally -oracle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: It is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1.
Abstract: Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G -independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge.
We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ℵ1. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA.
Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most ℵ1.
We refute various reasonable generalizations of these results to hypergraphs (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: It is shown that Player II has a winning strategy in the bound-countable game, thus establishing a new ZFC result and new proofs for two cardinal diamonds and the impossibility of collapsing cardinals to ℵ2 under certain conditions are given.
Abstract: We present results concerning winning strategies and tactics in club games on λ. We show that there is generally no winning tactic for the player trying to get inside the club. The bound-countable game turns out to be rather fruitful and adds to some previous results about the construction of elementary substructures and their localization in certain intervals. We show that Player II has a winning strategy in the bound-countable game, thus establishing a new ZFC result. The applications given are new proofs for two cardinal diamonds and the impossibility of collapsing cardinals to ℵ2 under certain conditions (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: This article provides a detailed comparison between two systems of collapsing functions, which play a crucial role in proof theory, in theAnalysis of patterns of resemblance, and the analysis of maximal order types of well partial orders.
Abstract: This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: It is proved that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to sentences cannot be extended to Πn + 2 sentences.
Abstract: We characterize the sets of all Π2 and all (= Boolean combinations of Σ1) theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: Using Koszmider's strongly unbounded functions, the following consistency result is shown: cardinal sequences of superatomic Boolean algebras can be cardinal sequences in some cardinal-preserving generic extension.
Abstract: Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra such that , for every α < η and (i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉η⌢〈λ〉). Especially, and can be cardinal sequences of superatomic Boolean algebras.
TL;DR: It is shown that the quantified propositional proof systems Gi are polynomially equivalent to their restricted versions that require all cut formulas to be prenex Σqi (orprenex Πqi).
Abstract: We show that the quantified propositional proof systems Gi are polynomially equivalent to their restricted versions that require all cut formulas to be prenex Σqi (or prenex Πqi). Previously this was known only for the treelike systems G*i. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and n-uniqueness for n-amalgamation problems of stable theories.
Abstract: In this paper, the connections between model theory and the theory of infinite permutation groups (see 11) are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n ⩾ 2, there exists a stable theory having (k + 1)-existence and k-uniqueness, for every k ⩽ n, but has neither (n + 2)-existence nor (n + 1)-uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: It is proved that the Nisan-Wigderson generators based on computationally hard functions and suitable matrices are hard for propositional proof systems that admit feasible interpolation.
Abstract: We prove that the Nisan-Wigderson generators based on computationally hard functions and suitable matrices are hard for propositional proof systems that admit feasible interpolation. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: It is shown that the representability of cylindric algebras by relativized set algeBRas depends on the scope of the operation transposition which can be defined on the algebra.
Abstract: We show that the representability of cylindric algebras by relativized set algebras depends on the scope of the operation transposition which can be defined on the algebra. The existence of “partial transposition” assures this kind of representability of the cylindric algebra (while the existence of transposition assures polyadic representation). Further we characterize those cylindric algebras in which the operator transposition can be introduced (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: A new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals is developed, using the number of normal measures carried by a measurable cardinal as an oracle.
Abstract: We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: Extensions with operations are considered, which internally represent description operators, unbounded set quantifiers and local fixed point operators, which turn out to be impredicative on the proof theoretic strength of the resulting systems.
Abstract: We study constructive set theories, which deal with (partial) operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be (except for the description operator) impredicative (being comparable with full second-order arithmetic and the second-order μ–calculus over arithmetic). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: The Craig interpolation theorem is shown for an extended LJ with strong negation and a new simple proof of this theorem is obtained.
Abstract: The Craig interpolation theorem is shown for an extended LJ with strong negation. A new simple proof of this theorem is obtained. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: It is proved that weakly measurable cardinals and measurableCardinals are equiconsistent, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal.
Abstract: In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in . Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: Several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity and are investigated for set-theoretic properties.
Abstract: We present several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set-theoretic properties of such σ-ideals.
TL;DR: It is proved definability of some probability concepts such as having F(u) as distribution function, independence and martingale property and Kolmogorov's existence theorem is deduced from the compactness theorem.
Abstract: We study model theory of random variables using finitary integral logic. We prove definability of some probability concepts such as having F(u) as distribution function, independence and martingale property. We then deduce Kolmogorov's existence theorem from the compactness theorem.
TL;DR: It is shown that rings over which (pure-) projectives are trivial are trivial in terms of various notions of separability of (flat) strict Mittag-Leffler modules.
Abstract: We characterize strict Mittag-Leffler modules in terms of free realizations of positive primitive formulas, and rings over which (pure-) projectives are trivial in terms of various notions of separability of (flat) strict Mittag-Leffler modules. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
TL;DR: This paper is a direct successor to 12 and introduces a new realisability interpretation for weak systems of explicit mathematics and uses it in order to analyze extensions of the theory PET in 12 by the so-called join axiom of explicit Mathematics.
Abstract: This paper is a direct successor to 12. Its aim is to introduce a new realisability interpretation for weak systems of explicit mathematics and use it in order to analyze extensions of the theory PET in 12 by the so-called join axiom of explicit mathematics.
TL;DR: It is shown that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located and if the subspace is, moreover, compact, then its collection of points is Bishop-compact.
Abstract: We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact.
TL;DR: The classic Rice and Rice-Shapiro Theorems for computably enumerable sets are extended to analogs for all the higher levels in the finite Ershov Hierarchy, and analogs in the transfinite Ershova Hierarchy are extended.
Abstract: Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, . Other cases are done for all transfinite notations in a very natural, proper subsystem of , where has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in .