Authors:Riccardo Camerlo; Jacques Duparc Pages: 195 - 201 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: 2018-05-01 DOI: 10.1007/s00153-017-0563-6 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Philip Scowcroft Pages: 239 - 272 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: 2018-05-01 DOI: 10.1007/s00153-017-0566-3 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Gunter Fuchs Pages: 273 - 284 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: 2018-05-01 DOI: 10.1007/s00153-017-0568-1 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Vladimir Kanovei; Vassily Lyubetsky Pages: 285 - 298 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: 2018-05-01 DOI: 10.1007/s00153-017-0569-0 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Sebastien Vasey Pages: 299 - 315 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: 2018-05-01 DOI: 10.1007/s00153-017-0570-7 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Hiroshi Sakai Pages: 317 - 327 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: 2018-05-01 DOI: 10.1007/s00153-017-0572-5 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Philipp Schlicht Pages: 329 - 359 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: 2018-05-01 DOI: 10.1007/s00153-017-0571-6 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Michel Marti; George Metcalfe Pages: 361 - 380 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: 2018-05-01 DOI: 10.1007/s00153-017-0573-4 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Lorenzo Carlucci Pages: 381 - 389 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: 2018-05-01 DOI: 10.1007/s00153-017-0576-1 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Petr Cintula; Carles Noguera Pages: 391 - 420 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: 2018-05-01 DOI: 10.1007/s00153-017-0577-0 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Jörg Brendle Pages: 421 - 428 Abstract: We show that, consistently, there can be maximal subtrees of \(\mathcal{P}(\omega )\) and \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) of arbitrary regular uncountable size below the size of the continuum \({\mathfrak c}\) . We also show that there are no maximal subtrees of \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) with countable levels. Our results answer several questions of Campero-Arena et al. (Fund Math 234:73–89, 2016). PubDate: 2018-05-01 DOI: 10.1007/s00153-017-0575-2 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Linda Lawton Pages: 429 - 451 Abstract: An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \((\mathscr {L}({\varPi }_1^0), \cap , \cup , 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure. PubDate: 2018-05-01 DOI: 10.1007/s00153-017-0564-5 Issue No:Vol. 57, No. 3-4 (2018)

Authors:Julia F. Knight; Vikram Saraph Pages: 453 - 472 Abstract: We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable \(\varSigma _3\) Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d- \(\varSigma _2\) ” (the conjunction of a computable \(\varSigma _2\) sentence and a computable \(\varPi _2\) sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of \(\mathbb {Q}\) . We show that for some of these groups, the computable \(\varSigma _3\) Scott sentence is best possible, while for others, there is a computable d- \(\varSigma _2\) Scott sentence. PubDate: 2018-05-01 DOI: 10.1007/s00153-017-0578-z Issue No:Vol. 57, No. 3-4 (2018)

Authors:Victoria Gitman; Joel David Hamkins Abstract: 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 \(\Delta _2\) -definable class containing no regular cardinals. In such a model, there can be no \(\Sigma _2\) -reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler. PubDate: 2018-05-14 DOI: 10.1007/s00153-018-0632-5

Authors:Luciano J. González Abstract: We define when a ternary term m of an algebraic language \(\mathcal {L}\) is called a distributive nearlattice term ( \(\mathrm {DN}\) -term) of a sentential logic \(\mathcal {S}\) . Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \(\mathrm {DN}\) -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 (under the point of view of Algebraic Logic) associated with a selfextensional logic with a \(\mathrm {DN}\) -term is a variety, and we obtain that the logic is in fact fully selfextensional. PubDate: 2018-05-11 DOI: 10.1007/s00153-018-0628-1

Authors:Stefano Baratella Abstract: We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):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 (2010), 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 consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem. PubDate: 2018-05-10 DOI: 10.1007/s00153-018-0630-7

Authors:Josef Berger; Hajime Ishihara; Takayuki Kihara; Takako Nemoto Abstract: We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ( \(\mathrm {BE}\) ) is equivalent to weak König lemma ( \(\mathrm {WKL}\) ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \(\mathrm {WKL}\) for convex trees, in the framework of constructive reverse mathematics. PubDate: 2018-05-10 DOI: 10.1007/s00153-018-0627-2

Authors:Rosalie Iemhoff Abstract: 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 \(\mathsf{K}\) . New is the result that \(\mathsf{KD}\) has uniform interpolation. The results imply that for modal logics \(\mathsf{K4}\) and \(\mathsf{S4}\) , which are known not to have uniform interpolation, certain sequent calculi cannot exist. PubDate: 2018-05-10 DOI: 10.1007/s00153-018-0629-0

Authors:Shehzad Ahmed Abstract: Shelah (Algorithms Comb 14:420–459, 1997) develops the theory of \(\mathrm {pcf}_I(A)\) without the assumption that \( A <\min (A)\) , going so far as to get generators for every \(\lambda \in \mathrm {pcf}_I(A)\) 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 \(\mathrm {pcf}_I(A)\) which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory. PubDate: 2018-05-09 DOI: 10.1007/s00153-018-0631-6

Authors:Marija Boričić Abstract: 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’ \(\Gamma \vdash _a^b\Delta \) with the intended meaning that “the probability of truthfulness of \(\Gamma \vdash \Delta \) belongs to the interval [a, b]”. This method makes it possible to define a system of derivations based on ’axioms’ of the form \(\Gamma _i\vdash _{a_i}^{b_i}\Delta _i\) , obtained as a result of empirical research, and then infer conclusions of the form \(\Gamma \vdash _a^b\Delta \) . We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus. PubDate: 2018-05-07 DOI: 10.1007/s00153-018-0626-3