location:  Publications → journals
Search results

Search: MSC category 03 ( Mathematical logic and foundations )

 Expand all        Collapse all Results 1 - 25 of 25

1. CJM Online first

Dow, Alan; Tall, Franklin D.
 Normality versus paracompactness in locally compact spaces This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$. Keywords:normal, paracompact, locally compact, countably tight, collectionwise Hausdorff, forcing with a coherent Souslin tree, Martin's Maximum, PFA(S)[S], Axiom R, moving off propertyCategories:54A35, 54D20, 54D45, 03E35, 03E50, 03E55, 03E57

2. CJM Online first

Harrison-Trainor, Matthew; Melnikov, Alexander; Miller, Russell
 On computable field embeddings and difference closed fields We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields. Keywords:computable algebra, algebraic field, difference field, extension of automorphismCategories:03D45, 03C57, 12Y05

3. CJM Online first

Speissegger, Patrick
 Quasianalytic Ilyashenko algebras I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields. Keywords:generalized series expansion, quasianalyticity, transition mapCategories:41A60, 30E15, 37D99, 03C99

4. CJM 2016 (vol 69 pp. 502)

Fischer, Vera; Mejia, Diego Alejandro
 Splitting, Bounding, and Almost Disjointness Can Be Quite Different We prove the consistency of $$\operatorname{add}(\mathcal{N})\lt \operatorname{cov}(\mathcal{N}) \lt \mathfrak{p}=\mathfrak{s} =\mathfrak{g}\lt \operatorname{add}(\mathcal{M}) = \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a} =\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c}$$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values. Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-namesCategories:03E17, 03E35, 03E40

5. CJM 2015 (vol 68 pp. 44)

Fernández Bretón, David J.
 Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property We answer two questions of Hindman, SteprÄns and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter. Keywords:ultrafilter, Stone-Cech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian groupCategories:03E75, 54D35, 54D80, 05D10, 05A18, 20K99

6. CJM 2015 (vol 68 pp. 675)

Martínez-de-la-Vega, Veronica; Mouron, Christopher
 Monotone Classes of Dendrites Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$. A continuum $X$ is isolated with respect to monotone maps if every continuum that is monotone equivalent to $X$ must also be homeomorphic to $X$. In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated. Keywords:dendrite, monotone, bqo, antichainCategories:54F50, 54C10, 06A07, 54F15, 54F65, 03E15

7. CJM 2013 (vol 66 pp. 759)

Hrušák, Michael; van Mill, Jan
 Addendum to "Nearly Countable Dense Homogeneous Spaces" This paper provides an addendum to M. HruÅ¡Ã¡k and J. van Mill Nearly countable dense homogeneous spaces.'' Canad. J. Math., published online 2013-03-08 http://dx.doi.org/10.4153/CJM-2013-006-8. Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjectureCategories:54H05, 03E15, 54E50

8. CJM 2013 (vol 66 pp. 903)

Sargsyan, Grigor; Trang, Nam
 Non-tame Mice from Tame Failures of the Unique Branch Hypothesis In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing $Ord \cup \mathbb{R}$ such that $M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this implies the existence (in $V$) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees. Keywords:mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBHCategories:03E15, 03E45, 03E60

9. CJM 2013 (vol 66 pp. 743)

Hrušák, Michael; van Mill, Jan
 Nearly Countable Dense Homogeneous Spaces We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n{-}1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\setminus S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets, is shown to be closely related to Topological Vaught's Conjecture. Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjectureCategories:54H05, 03E15, 54E50

10. CJM 2013 (vol 66 pp. 303)

Elekes, Márton; Steprāns, Juris
 Haar Null Sets and the Consistent Reflection of Non-meagreness A subset $X$ of a Polish group $G$ is called Haar null if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb R$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of BartoszyÅski and Burke-Miller. Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic realCategories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

11. CJM 2012 (vol 64 pp. 1378)

Raghavan, Dilip; Steprāns, Juris
 On Weakly Tight Families Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by HruÅ¡Ã¡k and GarcÃ­a Ferreira, who applied it to the KatÃ©tov order on almost disjoint families. Keywords:maximal almost disjoint family, cardinal invariantsCategories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

12. CJM 2012 (vol 65 pp. 222)

Sauer, N. W.
 Distance Sets of Urysohn Metric Spaces A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With $\operatorname{dist}(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous, universal, separable and complete. (It is not difficult to deduce that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds every separable metric space $\mathrm{M}$ with $\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.) The main results are: (1) A characterization of the sets $\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\mathrm{M}$ is homogeneous. Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stabilityCategories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99

13. CJM 2012 (vol 64 pp. 1182)

Tall, Franklin D.
 PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$ Extending the work of Larson and Todorcevic, we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$ is a coherent Souslin tree. Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinalCategories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65

14. CJM 2012 (vol 65 pp. 485)

Bice, Tristan Matthew
 Filters in C*-Algebras In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding. Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towersCategories:46L03, 03E35

15. CJM 2011 (vol 63 pp. 1416)

Shelah, Saharon
 MAD Saturated Families and SANE Player We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any two is finite) that is saturated (completely separable \emph{i.e.,} any $X \subseteq \mathbb{N}$ is included in a finite union of members of the family \emph{or} includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation: (i) if $2^{\aleph_0} \lt \aleph_\omega$, then there is such a family; (ii) if there is no such family, then some situation related to pcf holds whose consistency is large (and if ${\mathfrak a}_* \gt \aleph_1$ even unknown); (iii) if, \emph{e.g.,} there is no inner model with measurables, \emph{then} there is such a family. Keywords:set theory, MAD families, pcf, the continuumCategories:03E05, 03E04, 03E17

16. CJM 2010 (vol 62 pp. 481)

Casals-Ruiz, Montserrat; Kazachkov, Ilya V.
 Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable. Categories:20F10, 03C10, 20F06

17. CJM 2009 (vol 61 pp. 604)

Hart, Joan E.; Kunen, Kenneth
 First Countable Continua and Proper Forcing Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight $\aleph_1$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals. Keywords:connected space, Continuum Hypothesis, proper forcing, irreducible mapCategories:54D05, 03E35

18. CJM 2008 (vol 60 pp. 88)

 Nilpotent Conjugacy Classes in $p$-adic Lie Algebras: The Odd Orthogonal Case We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over $p$-adic fields definable? By definable, we mean definable by a formula in Pas's language. In this language, there are no field extensions and no uniformisers. Using Waldspurger's parametrization, we answer in the affirmative in the case of special orthogonal Lie algebras $\mathfrak{so}(n)$ for $n$ odd, over $p$-adic fields. Categories:17B10, 03C60

19. CJM 2007 (vol 59 pp. 575)

Hernández-Hernández, Fernando; Hrušák, Michael
 Cardinal Invariants of Analytic $P$-Ideals We study the cardinal invariants of analytic $P$-ideals, concentrating on the ideal $\mathcal{Z}$ of asymptotic density zero. Among other results we prove $\min\{ \mathfrak{b},\cov\ (\mathcal{N}) \} \leq\cov^{\ast}(\mathcal{Z}) \leq\max\{ \mathfrak{b},\non(\mathcal{N}) \right\}.$ Categories:03E17, 03E40

20. CJM 2006 (vol 58 pp. 768)

Hu, Zhiguo; Neufang, Matthias
 Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\m$. In this paper, we explore the close connection between $\dec(\m)$ and the cardinal level of the Mazur property for the predual $\m_*$ of $\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra $\lone$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $\luc^*$, etc. We show that for any of these von Neumann algebras, say $\m$, the cardinal number $\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kg$ of $G$ and the least cardinality $\bg$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $\ag^{**}$. Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centreCategories:22D05, 43A20, 43A30, 03E55, 46L10

21. CJM 2005 (vol 57 pp. 1139)

Burke, Maxim R.; Miller, Arnold W.
 Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set We prove that it is relatively consistent with $\ZFC$ that in any perfect Polish space, for every nonmeager set $A$ there exists a nowhere dense Cantor set $C$ such that $A\cap C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog. Keywords:Property of Baire, Lebesgue measure,, Cantor set, oracle forcingCategories:03E35, 03E17, 03E50

22. CJM 2005 (vol 57 pp. 471)

Ciesielski, Krzysztof; Pawlikowski, Janusz
 Small Coverings with Smooth Functions under the Covering Property Axiom In the paper we formulate a Covering Property Axiom, \psmP, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Stepr\={a}ns. \begin{compactenum}[\rm(a)~~] \item There exists a family $\F$ of less than continuum many $\C^1$ functions from $\real$ to $\real$ such that $\real^2$ is covered by functions from $\F$, in the sense that for every $\la x,y\ra\in\real^2$ there exists an $f\in\F$ such that either $f(x)=y$ or $f(y)=x$. \item For every Borel function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^1$'' functions ({\em i.e.,} differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$. \item For every $n>0$ and a $D^n$ function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^n$ functions whose graphs cover the graph of $f$. \end{compactenum} We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevski\v{\i}. Keywords:continuous, smooth, coveringCategories:26A24, 03E35

23. CJM 1999 (vol 51 pp. 309)

Leung, Denny H.; Tang, Wee-Kee
 Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$. Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12

24. CJM 1997 (vol 49 pp. 1089)

Burke, Maxim R.; Ciesielski, Krzysztof
 Sets on which measurable functions are determined by their range We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range. Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU)Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50

25. CJM 1997 (vol 49 pp. 641)

Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce
 Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law) Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81
 top of page | contact us | privacy | site map |