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Publisher: Springer-Verlag (Total: 2354 journals)

 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  [2354 journals]
• On the consistency strength of level by level inequivalence
• Authors: Arthur W. Apter
Pages: 715 - 723
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: 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: 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)

• Authors: David Asperó
Pages: 797 - 810
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: 2017-11-01
DOI: 10.1007/s00153-017-0539-6
Issue No: Vol. 56, No. 7-8 (2017)

• Equimorphy: the case of chains
• Authors: C. Laflamme; M. Pouzet; R. Woodrow
Pages: 811 - 829
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: 2017-11-01
DOI: 10.1007/s00153-017-0545-8
Issue No: Vol. 56, No. 7-8 (2017)

• Katětov order on Borel ideals
• Authors: Michael Hrušák
Pages: 831 - 847
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: 2017-11-01
DOI: 10.1007/s00153-017-0543-x
Issue No: Vol. 56, No. 7-8 (2017)

• On constructions with 2-cardinals
• Authors: Piotr Koszmider
Pages: 849 - 876
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)

• The nonstationary ideal on $$P_\kappa (\lambda )$$ P κ ( λ ) for
$$\lambda$$ λ singular
• Authors: Pierre Matet; Saharon Shelah
Pages: 911 - 934
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: 2017-11-01
DOI: 10.1007/s00153-017-0552-9
Issue No: Vol. 56, No. 7-8 (2017)

• The sharp for the Chang model is small
• Authors: William J. Mitchell
Pages: 935 - 982
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: 2017-11-01
DOI: 10.1007/s00153-017-0547-6
Issue No: Vol. 56, No. 7-8 (2017)

• Two applications of finite side conditions at $$\omega _2$$ ω 2
• Authors: Itay Neeman
Pages: 983 - 1036
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: 2017-11-01
DOI: 10.1007/s00153-017-0550-y
Issue No: Vol. 56, No. 7-8 (2017)

• Codings and strongly inaccessible cardinals
Pages: 1037 - 1044
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: 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)

• 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: 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: 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: 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: 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: 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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