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Quasi-uniform type spacesMar 15 2019In this article, we introduce and investigate the concept of partial quasi-metric type space as a generalization of both partial quasi-metric and quasi-metric type spaces. We show that many important constructions studied in K\"unzi's theory of partial ... More
Intrinsic Regular Surfaces of low co-dimension in Heisenberg groupsMar 11 2019In this paper we study intrinsic regular submanifolds of $\mathbb{H}^n$, of low co-dimension in relation with the regularity of their intrinsic parametrization. We extend some results proved for one co-dimensional $\mathbb{H}$-regular surfaces, characterizing ... More
Nilpotent Types and Fracture Squares in Homotopy Type TheoryMar 08 2019We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg-Mac Lane space are proven. We also construct ... More
Trace operator on von Koch's snowflakeMar 04 2019We study properties of the boundary trace operator on the Sobolev space $W^1_1(\Omega)$. Using density result by Koskela, Zhang we define a surjective operator \mbox{$Tr: W^1_1(\Omega_K)\rightarrow X(\Omega_K)$}, where $\Omega_K$ is a von Koch snowflake ... More
Capturing sets of ordinals by normal ultrapowersFeb 27 2019We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda)$ for $\lambda>\kappa$. We consider two versions of this questions, the capturing property $\mathrm{CP}(\kappa,\lambda)$ ... More
A lower density operator for the Borel algebraFeb 26 2019We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $\sigma$-ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.
Twisting $c_0$ around nonseparable Banach spacesFeb 20 2019We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis ... More
Set-theoretic aspects of accessible categoriesFeb 18 2019Mar 01 2019An accessible category is, roughly, a category with all sufficiently directed colimits, in which every object can be resolved as a directed system of "small" subobjects. Such categories admit a purely category-theoretic replacement for cardinality: the ... More
Set-theoretic aspects of accessible categoriesFeb 18 2019An accessible category is, roughly, a category with all sufficiently directed colimits, in which every object can be resolved as a directed system of "small" subobjects. Such categories admit a purely category-theoretic replacement for cardinality: the ... More
Homomorphisms Between Rings with Infinitesimals and Infinitesimal ComparisonsFeb 16 2019In this paper, we will examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's ... More
A $\square(κ)$-like principle consistent with weak compactnessFeb 11 2019Sun proved that when $\kappa$ is weakly compact, the \emph{$1$-club} subsets of $\kappa$ provide a filter base for the weakly compact ideal, and hence can also be used to give a characterization of weakly compact sets which resembles the definition of ... More
The method of forcingFeb 08 2019The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC.
The Cohomology of the Ordinals I: Basic Theory and Consistency ResultsFeb 07 2019In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is \v{C}ech cohomology. We show in particular that the \v{C}ech cohomology groups of the ordinals articulate ... More
The covering number of the strong measure zero ideal can be above almost everything elseFeb 05 2019We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates ... More
Categorical semantics of metric spaces and continuous logicJan 25 2019Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this "continuous semantics" is equivalent to the a priori ... More
A consistency result on long cardinal sequencesJan 25 2019For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$ it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f:\eta \to [\kappa,2^{\kappa}]\cap Card$ with $f(\alpha)=\kappa$ for $cf(\alpha)<\kappa$ is the cardinal sequence ... More
A consistency result on long cardinal sequencesJan 25 2019Feb 18 2019For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$ it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f:\eta \to [\kappa,2^{\kappa}]\cap Card$ with $f(\alpha)=\kappa$ for $cf(\alpha)<\kappa$ is the cardinal sequence ... More
On densely complete metric spaces and extensions of uniformly continuous functions in $\mathbf{ZF}$Jan 25 2019A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the countable axiom ... More
On Adjoint Additive ProcessesJan 11 2019Starting with an additive process $(Y_t)_{t\geq0}$, it is in certain cases possible to construct an adjoint process $(X_t)_{t\geq0}$ which is itself additive. Moreover, assuming that the transition densities of $(Y_t)_{t\geq0}$ are controlled by a natural ... More
Mechanization of Separation in Generic ExtensionsJan 10 2019We mechanize, in the proof assistant Isabelle, a proof of the axiom-scheme of Separation in generic extensions of models of set theory by using the fundamental theorems of forcing. We also formalize the satisfaction of the axioms of Extensionality, Foundation, ... More
The Dual Baer Criterion for non-perfect ringsJan 05 2019Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem (first formulated ... More
Dense ideals and cardinal arithmeticJan 04 2019From large cardinals we show the consistency of normal, fine, $\kappa$-complete $\lambda$-dense ideals on $\mathcal{P}_\kappa(\lambda)$ for successor $\kappa$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering ... More
Coherent forestsJan 04 2019A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ... More
On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb Z$ in ZFCDec 17 2018Following a recent paper by Herrlich and Tachtsis, we investigate in ZFC the following compactness question: for which unountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable ... More
On ultrafilter extensions of first-order models and ultrafilter interpretationsDec 15 2018There exist two known canonical concepts of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as its main ... More
Ways of DestructionDec 04 2018We study the following natural strong variant of destroying Borel ideals: $\mathbb{P}$ $\textit{$+$-destroys}$ $\mathcal{I}$ if $\mathbb{P}$ adds an $\mathcal{I}$-positive set which has finite intersection with every $A\in\mathcal{I}\cap V$. Also, we ... More
The Morris modelNov 27 2018Douglass B. Morris announced in 1970 of the following statement is consistent: "For every $\alpha$, there exists a set $A_\alpha$ which is the countable union of countable sets, and $\mathcal P(A_\alpha)$ can be partitioned into $\aleph_\alpha$ non-empty ... More
The Morris modelNov 27 2018Feb 13 2019Douglass B. Morris announced in 1970 that it is consistent with ZF that "For every $\alpha$, there exists a set $A_\alpha$ which is the countable union of countable sets, and $\mathcal P(A_\alpha)$ can be partitioned into $\aleph_\alpha$ non-empty sets". ... More
On intermediate extensions of generic extensions by a random realNov 26 2018The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that ... More
Voiculescu's Theorem for Nonseparable C*-algebrasNov 23 2018Nov 26 2018We prove that Voiculescu's noncommutative version of the Weyl-von Neumann theorem can be extended to all (not necessarily separable) unital, separably representable C*-algebras whose density character is strictly smaller than $\mathfrak{p}$. We show moreover ... More
A Sacks indestructible co-analytic maximal eventually different familyNov 14 2018In the constructible universe, we construct a co-analytic maximal family of pairwise eventually different functions from $\mathbb{N}$ to $\mathbb{N}$ which remains maximal after adding arbitrarily many Sacks reals (by a countably supported iteration or ... More
On a lattice-like property of quasi-arithmetic meansNov 12 2018We will prove that in a family of quasi-arithmetic means sattisfying certain smoothness assumption (embed with a naural pointwise ordering) every finite family has both supremum and infimum, which is also a quasi-arithmetic mean sattisfying the same smoothness ... More
All Parovichenko spaces are soft-ParovichenkoNov 09 2018Nov 14 2018It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $\mathfrak{c}$ is a remainder in a soft compactification of $\mathbb{N}$. We also exhibit an example of a compact space of weight~$\aleph_1$ --- hence a remainder ... More
Note on the definition of neutrosophic logicNov 07 2018Smarandache introduced a new logic called `neutrosophic logic'. Its definition contains many misuses of nonstandard analysis, and its description is entirely hand-waving. In this note, we describe a rigorous definition of neutrosophic logic and correct ... More
Characterizing large cardinals through Neeman's pure side condition forcingOct 31 2018We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial ... More
On Harrington's model in which Separation holds but Reduction fails at the 3rd projective level, and on some related models of SamiOct 30 2018Nov 11 2018In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbf\Pi^1_3$-Separation holds but $\mathbf\Sigma^1_3$-Reduction fails. The result has never appeared in a ... More
Existence of incompressible and immiscible flows in critical function spaces on bounded domainsOct 29 2018We study global existence and uniqueness of solutions to instationary inhomogeneous Navier-Stokes equations on bounded domains of $\R^n, n\geq 3$, with initial velocity in $B^0_{q,\infty}(\Om)$, $q\geq n$, and piecewise constant initial density. \par ... More
Inherent stochasticity precludes hysteresis in gene regulatory networksOct 24 2018Dec 18 2018Cell fate determination, the process through which cells commit to differentiated states, has been shown to be mediated by gene regulatory motifs with mutually exclusive expression states. The classical picture for deterministic cell decision making includes ... More
Ramsey subsets of the space of infinite block sequences of vectorsOct 23 2018We study families of infinite block sequences of elements of the space $\FIN_k$. In particular we study Ramsey properties of such families and Ramsey properties localized to a selective or semiselective coideal. We show how the stable ordered-union ultrafilters ... More
Stable ordered union ultrafilters and $\mathrm{cov}(\mathcal{M})<\mathfrak c$Oct 19 2018A union ultrafilter is an ultrafilter over the finite subsets of $\omega$ that has a base of sets of the form $\mathrm{FU}(X)$, where $X$ is an infinite pairwise disjoint family and $\mathrm{FU}(X)=\{\bigcup F\big|F\in[X]^{<\omega}\setminus\{\varnothing\}\}$. ... More
Existentially closed De Morgan algebrasOct 04 2018Oct 15 2018We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is $\aleph_0$-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for Boole-De Morgan algebras. ... More
Tameness in generalized metric structuresOct 04 2018Oct 15 2018We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) ... More
Hindman's finite sums theorem and its application to topologizations of algebrasOct 03 2018The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter ... More
Embedding $\mathrm{C}^*$-algebras into the Calkin algebraSep 29 2018Feb 25 2019Every $\mathrm{C}^*$-algebra, regardless of its density character, can be embedded into the Calkin algebra in a forcing extension of the universe obtained without collapsing any cardinal.
Embedding $\mathrm{C}^*$-algebras into the Calkin algebraSep 29 2018Jan 19 2019Every $\mathrm{C}^*$-algebra, regardless of its density character, can be embedded into the Calkin algebra in a forcing extension of the universe obtained without collapsing any cardinal.
A model with everything except for a well-ordering of the realsSep 27 2018We construct a model of $\mathsf{ZF} + \mathsf{DC}$ containing a Luzin set, a Sierpi\'{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.
Hausdorff Tight Groupoids GeneralisedSep 23 2018We extend Exel's ample tight groupoid construction to general locally compact \'etale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as 'pseudobases' of open bisections, thus yielding a duality which ... More
Knaster and friends I: Closed colorings and precalibersSep 22 2018Dec 04 2018The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares are not $\kappa$-cc ... More
Josefson-Nissenzweig property for $C_p$-spacesSep 19 2018The famous Rosenthal-Lacey theorem asserts that for each infinite compact space $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c_{0}$ or $\ell_{2}$. The aim of the paper is to study a natural variant of this result for the space ... More
Generalized Fuzzy metric Spaces with an application to Colour image filteringSep 17 2018Impulsive noise is a problem encountered during the acquisition and transmission of digital images. Fuzzy metrics dealing nicely with the nonlinear nature of digital images are used in vector median-based filters for noise reduction in colour and multichannel ... More
A category-theoretic characterization of almost measurable cardinalsSep 16 2018Oct 03 2018Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $\kappa$-chains, $\kappa$ a sufficiently large almost measurable cardinal. This condition on powerful ... More
Filter-linkedness and its effect on preservation of cardinal characteristicsSep 13 2018We introduce the property "$F$-linked" of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties "$\mu$-$F$-linked" and "$\theta$-$F$-Knaster" for posets in the natural way. We show that $\theta$-$F$-Knaster posets ... More
Computability at zero temperatureSep 01 2018Sep 07 2018In this paper, we investigate the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we prove that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable ... More
Combinatorics of Borel IdealsAug 28 2018In this work we study some combinatorial properties of Borel (or co-analytic) ideals on countable sets. We shall extend the theorem 4.18 presented in \cite{articulomaicol}, and also we will find an $F_\sigma$ tall ideal in which the player $II$ has a ... More
Parametrized Measuring and Club GuessingAug 26 2018Oct 20 2018We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\omega_1$ is measured by some club subset of $\omega_1$. The consistency ... More
On a conjecture for $\aleph_0$-bounded groupsAug 23 2018We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
Meager Sets, Games and Singular CardinalsAug 21 2018We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
Free sequences in P(ω)/finAug 17 2018We investigate maximal free sequences in the Boolean algebra $\mathcal{P}(\omega)/\mathrm{fin}$, as defined by D. Monk. We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality ... More
Halfway New Cardinal CharacteristicsAug 07 2018Nov 09 2018Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite ... More
Cichoń's Diagram and Localisation CardinalsAug 06 2018Aug 28 2018We reimplement the creature forcing construction used by Fischer et al. (arXiv:1402.0367) to separate Cicho\'{n}'s diagram into five cardinals as a countable support product. Using the fact that it is of countable support, we augment our construction ... More
Large irredundant sets in operator algebrasAug 04 2018Sep 02 2018A subset $\mathcal X$ of a C*-algebra $\mathcal A$ is called irredundant if no $A\in \mathcal X$ belongs to the C*-subalgebra of $\mathcal A$ generated by $\mathcal X\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and ... More
Set-Theoretic BlockchainsAug 04 2018Mar 05 2019Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while ... More
Set-Theoretic BlockchainsAug 04 2018Feb 12 2019Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while ... More
Characterizations of the weakly compact ideal on $P_κλ$Jul 31 2018Aug 23 2018Sun \cite{MR1245524} proved that a set $W\subseteq\kappa$ is $\Pi^1_1$-indescribable (or equivalently weakly compact) if and only if $W\cap C\neq\emptyset$ for every $1$-club $C\subseteq \kappa$; here, a set $C\subseteq\kappa$ is \emph{$1$-club} if and ... More
A note on groups definable in the p-adic fieldJul 24 2018It is known that a group G definable in the field of p-adic numbers is definably locally isomorphic to the group of Q_p-points of a connected algebraic group H defined over Q_p. We show that if H is commutative then G is commutative-by-finite. It follows ... More
First steps towards a formalization of ForcingJul 13 2018Nov 27 2018We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize the definition of forcing notions as preorders with ... More
Localization in Homotopy Type TheoryJul 11 2018We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order ... More
Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equationJul 03 2018In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient $A$ and initial conditions $X_0$ and $X_1$. In a previous study [Calbo G. et al, Comput. Math. Appl., 61(9), 2782--2792 (2011)], ... More
Products of Menger spaces in the Miller modelJun 27 2018Sep 21 2018We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between classical forcing posets with fusion and combinatorial ... More
Generalized solutions of variational problems and applicationsJun 27 2018Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study ... More
The Higher Cichoń DiagramJun 22 2018Aug 28 2018For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ... More
Dependent Choice, Properness, and Generic AbsolutenessJun 11 2018We observe that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals. We also investigate some basic consequences ... More
Ladder system uniformization on trees I & IIJun 11 2018Jan 04 2019Given a tree $T$ of height $\omega_1$, we say that a ladder system colouring $(f_\alpha)_{\alpha\in \lim\omega_1}$ has a $T$-uniformization if there is a function $\varphi$ defined on a subtree $S$ of $T$ so that for any $s\in S_\alpha$ of limit height ... More
Ilyashenko algebras based on transserial asymptotic expansionsJun 05 2018Jan 08 2019We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is ... More
The super tree property at the successor of a singularJun 03 2018For an inaccessible cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit ... More
Strictifying Homotopy Coherent Actions on Hochschild ComplexesMay 31 2018Dec 14 2018If P is a dg-operad acting on a dg-algebra A via algebra homomorphisms, then P acts on the Hochschild complex of A. In the more general case when P is a dg-prop, we show that P still acts on the Hochschild complex, but only up to coherent homotopy. We ... More
A note on non-classical Nonstandard ArithmeticMay 29 2018Nov 02 2018Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic, have been proposed. We single out one particular system by Dinis-Gaspar, which is categorised by the authors as being ... More
Many Different Uniformity Numbers of Yorioka IdealsMay 28 2018Feb 07 2019Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing ... More
Hausdorff compactifications in ZFMay 24 2018For a compactification $\alpha X$ of a Tychonoff space $X$, the algebra of all functions $f\in C(X)$ that are continuously extendable over $% \alpha X$ is denoted by $C_{\alpha}(X)$. It is shown that, in a model of $\textbf{ZF}$, it may happen that a ... More
A forcing axiom for a non-special Aronszajn treeMay 21 2018Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of ... More
Approximation of the probability density function of the randomized heat equation with non-homogeneous boundary conditionsMay 09 2018This paper deals with the randomized heat equation defined on a general bounded interval $[L_1,L_2]$ and with non-homogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic ... More
Critical CardinalsMay 07 2018We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is necessary for ... More
On the tightness of $G_δ$-modificationsMay 06 2018The $G_\delta$-modification $X_\delta$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_\delta$ subsets of $X$. Bella and Spadaro recently investigated the connection between ... More
Stationary ReflectionApr 30 2018We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.
Density of uniqueness triples from the diamond axiomApr 29 2018Nov 01 2018We work with a pre-$\lambda$-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $\lambda$. We investigate the ... More
On homeomorphisms and $C^{1}$ mapsApr 27 2018Our purpose in this article is first, following [8], to prove that if $\alpha $, $\beta $ are any points of the open unit disc $D(0;1)$ in the complex plane ${\bf C}$ and $r$, $s$ are any positive real numbers such that ${\overline{D}}( \alpha ;r) \subseteq ... More
$\leq_{SP}$ Can Have Infinitely Many ClassesApr 23 2018Apr 24 2018Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < \omega$. If we let $T_{n, ... More
The Fundamental Solution to the p-Laplacian in a class of Hörmander Vector FieldsApr 17 2018We find the fundamental solution to the p-Laplace equation in a class of H\"ormander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding ... More
Deeply concatenable subgroups might never be freeApr 16 2018We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness, completely metrizable ... More
On the resolvability of Lindelöf-generated and (countable extent)-generated spacesApr 09 2018Given a topological property $P$, we say that the space $X$ is $P$-generated if for any subset $A\subset X$ that is not open in $X$ there is a subspace $Y \subset X$ with property $P$ such that $A\cap Y$ is not open in $Y$. (Of course, in this definition ... More
An extension of an unicity class for Navier-Stokes equationsApr 09 2018This is a translation from French of my paper [R. May, Extension d'une classe d'unicite pour les equations de Navier-Stokes, Ann. I. H. Poincar\'{e}-AN 27 (2010) 705-718. doi:10.1016/j.anihp.2009.11.007]. Q. Chen, C. Miao, and Z. Zhang \cite{CMZ} have ... More
A footnote to The crisis in contemporary mathematicsApr 08 2018We examine the preparation and context of the paper "The Crisis in Contemporary Mathematics" by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation ... More
Every metric space is separable in function realizabilityApr 02 2018May 29 2018We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every discrete space is ... More
The Open Graph Axiom and Menger's ConjectureMar 22 2018Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $\sigma$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of Projective ... More
Products of $H$-separable spaces in the Laver modelMar 17 2018We prove that in the Laver model for the consistency of the Borel's conjecture, the product of any two $H$-separable spaces is $M$-separable.
Matrix iterations with vertical support restrictionsMar 14 2018Jul 12 2018We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on ... More
A model with Suslin trees but no minimal uncountable linear orders other than $ω_1$ and $-ω_1$Mar 09 2018Mar 12 2018We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $\omega_1$ and $-\omega_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration, proving that ... More
Completely Baire spaces, Menger spaces, and projective setsMar 09 2018W. Hurewicz proved that analytic Menger sets of reals are $\sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished ... More
Guessing models and the approachability idealFeb 27 2018Starting with two supercompact cardinals we produce a generic extension of the universe in which the principles ${\rm ISP}(\omega_2)$ and ${\rm ISP}(\omega_3)$ hold simultaneously, and the restriction of the approachability ideal $I[\omega_2]$ to the ... More
A Version of $κ$-Miller ForcingFeb 22 2018Mar 05 2019Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions the $\kappa$-Miller ... More