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 Subjects -> MATHEMATICS (Total: 909 journals)     - APPLIED MATHEMATICS (75 journals)    - GEOMETRY AND TOPOLOGY (20 journals)    - MATHEMATICS (676 journals)    - MATHEMATICS (GENERAL) (41 journals)    - NUMERICAL ANALYSIS (19 journals)    - PROBABILITIES AND MATH STATISTICS (78 journals) MATHEMATICS (676 journals)                  1 2 3 4 | Last

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 Archive for Mathematical Logic   [SJR: 0.946]   [H-I: 23]   [1 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0665 - ISSN (Online) 0933-5846    Published by Springer-Verlag  [2355 journals]
• On the consistency strength of level by level inequivalence
• Authors: Arthur W. Apter
Pages: 715 - 723
Abstract: 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: 2017-11-01
DOI: 10.1007/s00153-017-0538-7
Issue No: Vol. 56, No. 7-8 (2017)

• Strong measure zero and infinite games
• Authors: Fred Galvin; Jan Mycielski; Robert M. Solovay
Pages: 725 - 732
Abstract: 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: 2017-11-01
DOI: 10.1007/s00153-017-0541-z
Issue No: Vol. 56, No. 7-8 (2017)

• Same graph, different universe
• Authors: Assaf Rinot
Pages: 783 - 796
Abstract: 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: 2017-11-01
DOI: 10.1007/s00153-017-0551-x
Issue No: Vol. 56, No. 7-8 (2017)

• On constructions with 2-cardinals
• Authors: Piotr Koszmider
Pages: 849 - 876
Abstract: 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: 2017-11-01
DOI: 10.1007/s00153-017-0544-9
Issue No: Vol. 56, No. 7-8 (2017)

• Codings and strongly inaccessible cardinals
Pages: 1037 - 1044
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: 2017-11-01
DOI: 10.1007/s00153-017-0548-5
Issue No: Vol. 56, No. 7-8 (2017)

• Baumgartner’s isomorphism problem for $$\aleph _2$$ ℵ 2 -dense
suborders of $$\mathbb {R}$$ R
• Authors: Justin Tatch Moore; Stevo Todorcevic
Pages: 1105 - 1114
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: 2017-11-01
DOI: 10.1007/s00153-017-0549-4
Issue No: Vol. 56, No. 7-8 (2017)

• RETRACTED ARTICLE: A completeness theorem for continuous predicate modal
logic
• Authors: Stefano Baratella
Pages: 1135 - 1135
PubDate: 2017-11-01
DOI: 10.1007/s00153-017-0554-7
Issue No: Vol. 56, No. 7-8 (2017)

• Definability of types and VC density in differential topological fields
• Authors: Françoise Point
Abstract: Abstract Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.
PubDate: 2017-12-26
DOI: 10.1007/s00153-017-0607-y

• Homomorphism reductions on Polish groups
• Authors: Konstantinos A. Beros
Abstract: Abstract In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and $$H,L \subseteq G$$ are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism $$\varphi : G \rightarrow G$$ such that $$H = \varphi ^{-1} (L)$$ . We previously showed that there is a $$K_\sigma$$ subgroup L of the countable power of any locally compact Polish group G such that every $$K_\sigma$$ subgroup of $$G^\omega$$ is homomorphism reducible to L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.
PubDate: 2017-12-15
DOI: 10.1007/s00153-017-0606-z

• Model completeness of generic graphs in rational cases
• Authors: Hirotaka Kikyo
Abstract: Abstract Let $$\mathbf {K}_f$$ be an ab initio amalgamation class with an unbounded increasing concave function f. We show that if the predimension function has a rational coefficient and f satisfies a certain assumption then the generic structure of $$\mathbf {K}_f$$ has a model complete theory.
PubDate: 2017-12-13
DOI: 10.1007/s00153-017-0601-4

• Embedding locales and formal topologies into positive topologies
• Authors: Francesco Ciraulo; Giovanni Sambin
Abstract: Abstract A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a reflective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales.
PubDate: 2017-12-07
DOI: 10.1007/s00153-017-0605-0

• A partition relation for pairs on $$\omega ^{\omega ^\omega }$$ ω ω ω
• Authors: Claribet Piña
Abstract: Abstract We consider colorings of the pairs of a family $$\mathcal {F}\subseteq {{\mathrm{FIN}}}$$ of topological type $$\omega ^{\omega ^k}$$ , for $$k>1$$ ; and we find a homogeneous family $$\mathcal {G}\subseteq \mathcal {F}$$ for each coloring. As a consequence, we complete our study of the partition relation $${\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}$$ identifying $$\omega ^{\omega ^\omega }$$ as the smallest ordinal space $$\alpha <\omega _1$$ satisfying $${\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}$$ .
PubDate: 2017-12-04
DOI: 10.1007/s00153-017-0604-1

• On a class of maximality principles
• Authors: Daisuke Ikegami; Nam Trang
Abstract: Abstract We study various classes of maximality principles, $$\mathrm {MP}(\kappa ,\Gamma )$$ , introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where $$\Gamma$$ defines a class of forcing posets and $$\kappa$$ is an infinite cardinal. We explore the consistency strength and the relationship of $$\textsf {MP}(\kappa ,\Gamma )$$ with various forcing axioms when $$\kappa \in \{\omega ,\omega _1\}$$ . In particular, we give a characterization of bounded forcing axioms for a class of forcings $$\Gamma$$ in terms of maximality principles MP $$(\omega _1,\Gamma )$$ for $$\Sigma _1$$ formulas. A significant part of the paper is devoted to studying the principle MP $$(\kappa ,\Gamma )$$ where $$\kappa \in \{\omega ,\omega _1\}$$ and $$\Gamma$$ defines the class of stationary set preserving forcings. We show that MP $$(\kappa ,\Gamma )$$ has high consistency strength; on the other hand, if $$\Gamma$$ defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP $$(\kappa ,\Gamma )$$ is consistent relative to $$V=L$$ .
PubDate: 2017-11-27
DOI: 10.1007/s00153-017-0603-2

• Good frames in the Hart–Shelah example
• Authors: Will Boney; Sebastien Vasey
Abstract: For a fixed natural number $$n \ge 1$$ , the Hart–Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to $$\aleph _{n}$$ . We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good $$\aleph _{n - 1}$$ -frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the way, we develop new tools to build and analyze good frames.
PubDate: 2017-11-21
DOI: 10.1007/s00153-017-0599-7

• On the minimal cover property and certain notions of finite
• Authors: Eleftherios Tachtsis
Abstract: Abstract In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.
PubDate: 2017-11-17
DOI: 10.1007/s00153-017-0595-y

• Derivatives of normal functions and $$\omega$$ ω -models
• Authors: Toshiyasu Arai
Abstract: Abstract In this note the well-ordering principle for the derivative $$\mathsf{g}^{\prime }$$ of normal functions $$\mathsf{g}$$ on ordinals is shown to be equivalent to the existence of arbitrarily large countable coded $$\omega$$ -models of the well-ordering principle for the function $$\mathsf{g}$$ .
PubDate: 2017-11-15
DOI: 10.1007/s00153-017-0600-5

• Ax–Schanuel for linear differential equations
• Authors: Vahagn Aslanyan
Abstract: Abstract We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.
PubDate: 2017-11-14
DOI: 10.1007/s00153-017-0602-3

• Borel globalizations of partial actions of Polish groups
• Authors: H. Pinedo; C. Uzcategui
Abstract: Abstract We show that the enveloping space $${\mathbb {X}}_G$$ of a partial action of a Polish group G on a Polish space $${\mathbb {X}}$$ is a standard Borel space, that is to say, there is a topology $$\tau$$ on $${\mathbb {X}}_G$$ such that $$({\mathbb {X}}_G, \tau )$$ is Polish and the quotient Borel structure on $${\mathbb {X}}_G$$ is equal to $$Borel({\mathbb {X}}_G,\tau )$$ . To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught’s transform are valid for partial actions of groups.
PubDate: 2017-11-02
DOI: 10.1007/s00153-017-0598-8

• Strongly uplifting cardinals and the boldface resurrection axioms
• Authors: Joel David Hamkins; Thomas A. Johnstone
Abstract: Abstract We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
PubDate: 2017-08-19
DOI: 10.1007/s00153-017-0542-y

• Dedicated to James Earl Baumgartner
• PubDate: 2017-07-21
DOI: 10.1007/s00153-017-0574-3

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