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

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

Authors:Gunter Fuchs 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: 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: 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: 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: 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: 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: 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: 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: 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: 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

Authors:Sam Buss Abstract: We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with \(\hbox {MOD}_{m}\) gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of \(\hbox {AC}^{0}[p]\) circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed. PubDate: 2017-05-26 DOI: 10.1007/s00153-017-0560-9

Authors:Paula Henk; Albert Visser Abstract: This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic \(\mathsf {ILM}\) of Peano Arithmetic ( \(\mathsf {PA}\) ). It is well-known that any theories extending \(\mathsf {PA}\) have a supremum in the interpretability ordering. While provable in \(\mathsf {PA}\) , this fact is not reflected in the theorems of the modal system \(\mathsf {ILM}\) , due to limited expressive power. Our goal is to enrich the language of \(\mathsf {ILM}\) by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic \(\mathsf {GL}\) . PubDate: 2017-05-26 DOI: 10.1007/s00153-017-0557-4

Authors:William J. Mitchell Abstract: Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \(\kappa \) having an extender of length \(\kappa ^{+\omega _1}\) . PubDate: 2017-05-25 DOI: 10.1007/s00153-017-0547-6

Authors:Stefano Baratella Abstract: We study a predicate extension of an unbounded real valued propositional logic that has been recently introduced. The latter, in turn, can be regarded as an extension of both the abelian logic and of the propositional continuous logic. Among other results, we prove that our predicate extension satisfies the property of weak completeness (the equivalence between satisfiability and consistency) and, under an additional assumption on the set of premisses, the property of strong completeness (the equivalence between logical consequence and provability). Eventually we discuss some topological properties of the space of types in our logic. PubDate: 2017-05-25 DOI: 10.1007/s00153-017-0558-3

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