Authors:Yinhe Peng Pages: 237 - 251 Abstract: First, we show that every coherent tree that contains a Countryman suborder is \({\mathbb {R}}\) -embeddable when restricted to a club. Then for a linear order O that can not be embedded into \(\omega \) , there exists (consistently) an \({{\mathbb {R}}}\) -embeddable O-ranging coherent tree which is not Countryman. And for a linear order \(O'\) that can not be embedded into \({\mathbb {Z}}\) , there exists (consistently) an \({\mathbb {R}}\) -embeddable \(O'\) -ranging coherent tree which contains no Countryman suborder. Finally, we will see that this is the best we can do. PubDate: 2017-05-01 DOI: 10.1007/s00153-017-0530-2 Issue No:Vol. 56, No. 3-4 (2017)

Authors:Somayyeh Tari Pages: 309 - 317 Abstract: Let \({\mathcal {M}}=(M,<,+,\cdot ,\ldots )\) be a non-valuational weakly o-minimal expansion of a real closed field \((M,<,+,\cdot )\) . In this paper, we prove that \({\mathcal {M}}\) has a \(C^r\) -strong cell decomposition property, for each positive integer r, a best analogous result from Tanaka and Kawakami (Far East J Math Sci (FJMS) 25(3):417–431, 2007). We also show that curve selection property holds in non-valuational weakly o-minimal expansions of ordered groups. Finally, we extend the notion of definable compactness suitable for weakly o-minimal structures which was examined for definable sets (Peterzil and Steinhorn in J Lond Math Soc 295:769–786, 1999), and prove that a definable set is definably compact if and only if it is closed and bounded. PubDate: 2017-05-01 DOI: 10.1007/s00153-017-0523-1 Issue No:Vol. 56, No. 3-4 (2017)

Authors:Pierre Matet; Saharon Shelah Abstract: We give a new characterization of the nonstationary ideal on \(P_\kappa (\lambda )\) in the case when \(\kappa \) is a regular uncountable cardinal and \(\lambda \) a singular strong limit cardinal of cofinality at least \(\kappa \) . PubDate: 2017-05-19 DOI: 10.1007/s00153-017-0552-9

Authors:Fernando Ferreira; Gilda Ferreira Abstract: We introduce a new typed combinatory calculus with a type constructor that, to each type \(\sigma \) , associates the star type \(\sigma ^*\) of the nonempty finite subsets of elements of type \(\sigma \) . We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements. PubDate: 2017-05-19 DOI: 10.1007/s00153-017-0555-6

Authors:J. A. Larson Abstract: James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. PubDate: 2017-05-18 DOI: 10.1007/s00153-017-0546-7

Authors:James H. Schmerl Abstract: Suppose that \({\mathcal M}\models \mathsf{PA}\) and \({\mathfrak X} \subseteq {\mathcal P}(M)\) . If \({\mathcal M}\) has a finitely generated elementary end extension \({\mathcal N}\succ _\mathsf{end} {\mathcal M}\) such that \(\{X \cap M : X \in {{\mathrm{Def}}}({\mathcal N})\} = {\mathfrak X}\) , then there is such an \({\mathcal N}\) that is, in addition, a minimal extension of \({\mathcal M}\) iff every subset of M that is \(\Pi _1^0\) -definable in \(({\mathcal M}, {\mathfrak X})\) is the countable union of \(\Sigma _1^0\) -definable sets. PubDate: 2017-05-15 DOI: 10.1007/s00153-017-0556-5

Authors:Piotr Koszmider Abstract: We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper (Baumgartner and Shelah in Ann Pure Appl Logic 33(2):109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \(f:[\lambda ^{++}]^2\rightarrow [\lambda ^{++}]^{\le \lambda }\) with the appropriate \(\lambda ^+\) -version of property \(\Delta \) for regular \(\lambda \ge \omega \) satisfying \(\lambda ^{<\lambda }=\lambda \) . PubDate: 2017-05-15 DOI: 10.1007/s00153-017-0544-9

Authors:Michael Hrušák Abstract: We study the Katětov order on Borel ideals. We prove two structural theorems (dichotomies), one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals. PubDate: 2017-05-10 DOI: 10.1007/s00153-017-0543-x

Authors:C. Laflamme; M. Pouzet; R. Woodrow Abstract: Two structures are said to be equimorphic if each embeds in the other. Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case of linear orders, here also called chains. In particular we provide structure results for chains having less than continuum many isomorphism classes of equimorphic chains. We deduce as a corollary that any chain has either a single isomorphism class of equimorphic chains or infinitely many. PubDate: 2017-05-10 DOI: 10.1007/s00153-017-0545-8

Authors:David Asperó Abstract: I define a homogeneous \(\aleph _2\) –c.c. proper product forcing for adding many clubs of \(\omega _1\) with finite conditions. I use this forcing to build models of \(\mathfrak {b}(\omega _1)=\aleph _2\) , together with \(\mathfrak {d}(\omega _1)\) and \(2^{\aleph _0}\) large and with very strong failures of club guessing at \(\omega _1\) . PubDate: 2017-05-09 DOI: 10.1007/s00153-017-0539-6

Authors:Fred Galvin; Jan Mycielski; Robert M. Solovay Abstract: We show that strong measure zero sets (in a \(\sigma \) -totally bounded metric space) can be characterized by the nonexistence of a winning strategy in a certain infinite game. We use this characterization to give a proof of the well known fact, originally conjectured by K. Prikry, that every dense \(G_\delta \) subset of the real line contains a translate of every strong measure zero set. We also derive a related result which answers a question of J. Fickett. PubDate: 2017-05-08 DOI: 10.1007/s00153-017-0541-z

Authors:Assaf Rinot Abstract: May the same graph admit two different chromatic numbers in two different universes? How about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal \(\mu \) below the first fixed-point of the \(\aleph \) -function, there exists a graph \(\mathcal G_\mu \) satisfying the following: \(\mathcal G_\mu \) has size and chromatic number \(\mu \) ; for every infinite cardinal \(\kappa <\mu \) , there exists a cofinality-preserving \({{\mathrm{GCH}}}\) -preserving forcing extension in which \({{\mathrm{Chr}}}(\mathcal G_\mu )=\kappa \) . PubDate: 2017-05-08 DOI: 10.1007/s00153-017-0551-x

Authors:Natasha Dobrinen; José G. Mijares; Timothy Trujillo Abstract: A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not \(k+1\) -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \(\mathcal {P}(n)\) . If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \([\omega ]^{<\omega }\) . In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. PubDate: 2017-05-08 DOI: 10.1007/s00153-017-0540-0

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 [3], 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: 2017-05-04 DOI: 10.1007/s00153-017-0554-7

Authors:Arthur W. Apter Abstract: We show that the theories “ZFC \(+\) There is a supercompact cardinal” and “ZFC \(+\) There is a supercompact cardinal \(+\) Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent. PubDate: 2017-05-02 DOI: 10.1007/s00153-017-0538-7

Authors:Daniel W. Cunningham Abstract: Assuming \(\text {ZF}+\text {DC}\) , we prove that if there exists a strong partition cardinal greater than \(\varTheta \) , then (1) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ {{{\mathbb {R}}} }^{{\#}}\) exists, and (2) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ (\exists \kappa >\varTheta )\,(\kappa \) is measurable). Here \(\varTheta \) is the supremum of the ordinals which are the surjective image of the set of reals \({{{\mathbb {R}}} }\) . PubDate: 2017-03-18 DOI: 10.1007/s00153-017-0529-8

Authors:Joël Adler Abstract: As the class \(\mathcal {PCSL}\) of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a \(\aleph _0\) -categorical model companion \(\mathcal {PCSL}^*\) . As \(\mathcal {PCSL}\) is inductive the models of \(\mathcal {PCSL}^*\) are exactly the existentially closed models of \(\mathcal {PCSL}\) . We will construct the unique existentially closed countable model of \(\mathcal {PCSL}\) as a direct limit of algebraically closed pseudocomplemented semilattices. PubDate: 2017-03-17 DOI: 10.1007/s00153-017-0527-x

Authors:Mohammad Golshani Abstract: We show that Shelah cardinals are preserved under the canonical \({{\mathrm{GCH}}}\) forcing notion. We also show that if \({{\mathrm{GCH}}}\) holds and \(F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}\) is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies \(\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )\) . This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5–6):569–591, 2013) and independently by Honzik (Acta Univ Carol 1:55–72, 2015). We also prove an indestructibility result for Shelah cardinals. PubDate: 2017-03-07 DOI: 10.1007/s00153-017-0528-9

Authors:Takashi Sato; Takeshi Yamazaki Abstract: The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within \(\mathsf {RCA_0}\) give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over \(\mathsf {RCA_0}\) . Then we within \(\mathsf {RCA_0}\) give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over \(\mathsf {RCA_0}\) . Here the converses state that some fixed point properties characterize the completeness of the underlying spaces. PubDate: 2017-02-27 DOI: 10.1007/s00153-017-0526-y

Authors:Tapani Hyttinen; Vadim Kulikov; Miguel Moreno Abstract: We study the \(\kappa \) -Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and \(T^{\prime }\) , if T is classifiable and \(T^{\prime }\) is not, then the isomorphism of models of \(T^{\prime }\) is strictly above the isomorphism of models of T with respect to \(\kappa \) -Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions. PubDate: 2017-02-02 DOI: 10.1007/s00153-017-0521-3