Authors:Petr Cintula; Carles Noguera Abstract: This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers (Cintula and Noguera in Arch Math Logic 49(4):417–446, 2010; Arch Math Logic 53(3):353–372, 2016). PubDate: 2017-07-31 DOI: 10.1007/s00153-017-0577-0

Authors:Lorenzo Carlucci Abstract: Abstract Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements. PubDate: 2017-07-24 DOI: 10.1007/s00153-017-0576-1

Authors:Michel Marti; George Metcalfe Abstract: Abstract We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and (appropriately defined) modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics. PubDate: 2017-07-08 DOI: 10.1007/s00153-017-0573-4

Authors:Philipp Schlicht Abstract: Abstract If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\) . As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\) , is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs. PubDate: 2017-07-07 DOI: 10.1007/s00153-017-0571-6

Authors:Hiroshi Sakai Abstract: Abstract Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following: The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders. The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders. In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\) . PubDate: 2017-06-29 DOI: 10.1007/s00153-017-0572-5

Authors:Sebastien Vasey Abstract: We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim–Skolem–Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber’s quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber’s definition of a quasiminimal pregeometry class. PubDate: 2017-06-28 DOI: 10.1007/s00153-017-0570-7

Authors:Vladimir Kanovei; Vassily Lyubetsky Abstract: Abstract It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \(\text {OD}\) elements. PubDate: 2017-06-24 DOI: 10.1007/s00153-017-0569-0

Authors:Benno van den Berg; Eyvind Briseid; Pavol Safarik Abstract: Abstract In earlier work we introduced two systems for nonstandard analysis, one based on classical and one based on intuitionistic logic; these systems were conservative extensions of first-order Peano and Heyting arithmetic, respectively. In this paper we study how adding the principle of countable saturation to these systems affects their proof-theoretic strength. We will show that adding countable saturation to our intuitionistic system does not increase its proof-theoretic strength, while adding it to the classical system increases the strength from first- to full second-order arithmetic. PubDate: 2017-06-21 DOI: 10.1007/s00153-017-0567-2

Authors:Gunter Fuchs Abstract: Abstract It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length \(\omega _1\) sequence of normal ultrafilters, increasing in the Mitchell order, to \(\omega _1\) , is subcomplete. PubDate: 2017-06-20 DOI: 10.1007/s00153-017-0568-1

Authors:Sy-David Friedman; Giorgio Laguzzi Abstract: Abstract In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of \(2^\kappa \) , \(\kappa \) inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating \(\_\) lambda \(\,>\,\) cov \(\_\lambda \) (meagre)), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is \(\kappa ^\kappa \) -bounding, \(<\kappa \) -closed and \(\kappa ^+\) -cc, for \(\kappa \) inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362) provide a proof for (Shelah, On CON(Dominating \(\_\) lambda \(\,>\,\) cov \(\_\lambda \) (meagre)), arXiv:0904.0817, Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper (Khomskii et al. in Math L Q 62(4–5):439–456, 2016). PubDate: 2017-06-19 DOI: 10.1007/s00153-017-0562-7

Authors:Philip Scowcroft Abstract: Abstract In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\) ; shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\) -categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\) ; shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas. PubDate: 2017-06-15 DOI: 10.1007/s00153-017-0566-3

Authors:Haydar Göral Abstract: Abstract In this paper, we study an algebraically closed field \(\Omega \) expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \(\Gamma \) . This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \((\Omega , k, \Gamma )\) . This enables us to characterize the interpretable groups when \(\Gamma \) is divisible. Every interpretable group H in \((\Omega ,k, \Gamma )\) is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \) by an interpretable group N, which is the quotient of an algebraic group by a subgroup \(N_1\) , which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \) . PubDate: 2017-06-07 DOI: 10.1007/s00153-017-0565-4

Authors:Justin Tatch Moore; Stevo Todorcevic Abstract: Abstract In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \(2^{\aleph _0} \ge \aleph _2\) and every pair of \(\aleph _2\) -dense subsets of \(\mathbb {R}\) are isomorphic as linear orders. The main result is the isolation of a combinatorial principle \((**)\) which is immune to c.c.c. forcing and which in the presence of \(2^{\aleph _0} \le \aleph _2\) implies that two \(\aleph _2\) -dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC that there exists an \(\aleph _2\) dense suborder X of \(\mathbb {R}\) which cannot be embedded into \(-X\) in any outer model with the same \(\aleph _2\) . PubDate: 2017-06-06 DOI: 10.1007/s00153-017-0549-4

Authors:Arthur Fischer; Martin Goldstern; Jakob Kellner; Saharon Shelah Abstract: Abstract We use a (countable support) creature construction to show that consistently $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal M)< {{\mathrm{non}}}(\mathcal N)< {{\mathrm{cof}}}(\mathcal N) < 2^{\aleph _0}. \end{aligned}$$ The same method shows the consistency of $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal M)< {{\mathrm{cof}}}(\mathcal N) < 2^{\aleph _0}. \end{aligned}$$ PubDate: 2017-06-03 DOI: 10.1007/s00153-017-0553-8

Authors:Michael Rathjen; Jeroen Van der Meeren; Andreas Weiermann Abstract: Abstract In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than \(\varepsilon _0\) . We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. PubDate: 2017-06-02 DOI: 10.1007/s00153-017-0559-2

Authors:Riccardo Camerlo; Jacques Duparc Abstract: Abstract We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to \( \mathbb N^{\mathbb N}\) that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class. PubDate: 2017-06-01 DOI: 10.1007/s00153-017-0563-6

Authors:Tadatoshi Miyamoto Abstract: Abstract We show that a coding principle introduced by J. Moore with respect to all ladder systems is equiconsistent with the existence of a strongly inaccessible cardinal. We also show that a coding principle introduced by S. Todorcevic has consistency strength at least of a strongly inaccessible cardinal. PubDate: 2017-05-29 DOI: 10.1007/s00153-017-0548-5

Authors:Itay Neeman Abstract: Abstract We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \(\omega _2\) . The first involves adding a \(\Box _{\omega _1}\) sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \(\omega _2\) . PubDate: 2017-05-27 DOI: 10.1007/s00153-017-0550-y

Authors:Sebastien Vasey Abstract: Theorem 0.1 Let \(\mathbf {K}\) be an abstract elementary class (AEC) with amalgamation and no maximal models. Let \(\lambda > {LS}(\mathbf {K})\) . If \(\mathbf {K}\) is categorical in \(\lambda \) , then the model of cardinality \(\lambda \) is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: \(\mathbf {K}\) has a unique limit model in each cardinal below \(\lambda \) , (when \(\lambda \) is big-enough) \(\mathbf {K}\) is weakly tame below \(\lambda \) , and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by Shelah): Corollary 0.2 Let \(\mathbf {K}\) be an AEC with amalgamation and no maximal models. Let \(\lambda> \mu > {LS}(\mathbf {K})\) . If \(\mathbf {K}\) is solvable in \(\lambda \) , then \(\mathbf {K}\) is solvable in \(\mu \) . PubDate: 2017-05-27 DOI: 10.1007/s00153-017-0561-8