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Profiniteness in finitely generated varieties is undecidableDec 25 2017Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety $\mathcal V$ is standard if every Boolean topological algebra with the algebraic reduct ... More

A generic property of Solovay's set $Σ$Nov 01 2016We prove that Solovay's set $\Sigma$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.

Weighted sub-Laplacians on Métivier Groups: Essential Self-Adjointness and SpectrumOct 31 2016Let $G$ be a M\'etivier group and let $N$ be any homogeneous norm on $G$. For $\alpha>0$ denote by $w_\alpha$ the function $e^{-N^\alpha}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_\alpha}$ associated with the Dirichlet form $\phi \mapsto ... More

Bootstrapping structural properties, via accessible imagesOct 25 2016We present several new model-theoretic applications of the fact that, under a mild large cardinal assumption, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the results ... More

Bootstrapping structural properties, via accessible imagesOct 25 2016Nov 01 2016We present several new model-theoretic applications of the fact that, under a mild large cardinal assumption, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the results ... More

On subgroups of first homologyOct 18 2016We prove several new theorems regarding first homology. Some dichotomies for first homology of Peano continua are presented, as well as a notion of strong abelianization for arbitrary path connected metric spaces. We also show that the fundamental group ... More

Fodor's lemma can fail everywhereOct 13 2016We show that it is equiconsistent with $\mathsf{ZF}$ that Fodor's lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even $\sigma$-complete. Moreover, these failures can be controlled in a very precise manner. ... More

Definable discrete sets with large continuumOct 11 2016Let $\mathcal R$ be a $\Sigma^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a $\Delta^1_2$ maximal ... More

Special Aronszajn Tree PropertyOct 05 2016Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $\kappa$, there are $\kappa^+$-Aronszajn trees and all such trees are special.

Model error moment estimation via data assimilationOct 04 2016Using a dynamical model to make predictions about a system has many sources of error. These can include errors in how the model was initialised but also errors in the dynamics of the model itself. For many applications in data assimilation, probabilistic ... More

Namba forcing, weak approximation, and guessingOct 02 2016We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^\omega \le \omega_2$ is consistent with the existence of a nontrivial $\omega_1$-distributive ... More

Locally Compact Stone DualitySep 30 2016Oct 26 2016We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices ... More

Locally Compact Stone DualitySep 30 2016We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices ... More

Generic I0 at $\aleph_ω$Sep 26 2016In this paper it is introduced a generic large cardinal akin to I0, and its consequences are analyzed in the case that $\aleph_\omega$ is such a generic large cardinal. In this case $\aleph_\omega$ is J\'{o}nsson, and in a choiceless inner model many ... More

Coherent systems of finite support iterationsSep 18 2016We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cicho\'n's ... More

On the class of perfectly null sets and its transitive versionSep 13 2016We introduce two new classes of special subsets of the real line: the class of perfectly null sets and the class of sets which are perfectly null in the transitive sense. These classes may play the role of duals to the corresponding classes on the category ... More

A 1-separably injective space that does not contain $\ell_\infty$Sep 09 2016Nov 24 2016We show that the problem whether every $1$-separably injective Banach space contains an isomorphic copy of $\ell_\infty$ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the continuum hypothesis, ... More

A 1-separably injective space that does not contain $\ell_\infty$Sep 09 2016We study the $\omega_2$-subsets of tightly $\sigma$-filtered Boolean algebras and, as an application, we show the consistency of the existence of a Banach space that is 1-separably injective but does not contain any isomorphic copy of $\ell_\infty$.

Countable OD sets of reals belong to the ground modelSep 05 2016It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension that every countable ordinal-definable set of reals belongs to to the ground universe

Countable OD sets of reals belong to the ground modelSep 05 2016Oct 28 2016It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension that every countable ordinal-definable set of reals belongs to to the ground universe

Countable OD sets of reals belong to the ground modelSep 05 2016Nov 20 2016It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension that every countable ordinal-definable set of reals belongs to to the ground universe

From geometry to geology: An invitation to mathematical pluralism through the phenomenon of independenceSep 01 2016This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this universe that ... More

Biharmonic functions on the special unitary group SU(2)Aug 31 2016We construct new proper biharmonic functions defined on open and dense subsets of the special unitary group SU(2). Then we employ a duality principle to obtain new proper biharmonic functions from the non-compact 3-dimensional hyperbolic space H^3.

Biharmonic functions on the classical compact simple Lie groupsAug 30 2016The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $\SU n$, $\SO n$ and $\Sp n$. We work in a geometric setting which connects our study with ... More

Discrete Subsets in Topological Groups and Countable Extremally Disconnected GroupsAug 11 2016Sep 01 2016It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on the existence ... More

Strong failures of higher analogs of Hindman's theoremAug 04 2016Nov 22 2016We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with ... More

Strong failures of higher analogs of Hindman's theoremAug 04 2016Sep 24 2016We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There is a proper class of uncountable cardinals $\kappa$ satisfying the following statement: For every commutative ... More

Supercompact Extender Based Magidor-Radin ForcingAug 01 2016The extender based Magidor-Radin forcing is being generalized to supercompact type extenders.

Some applications of Supercompact Extender Based Forcings to HODAug 01 2016Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.

A spectral theory for simply periodic solutions of the sinh-Gordon equationJul 29 2016In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described ... More

Topological spaces with an $ω^ω$-baseJul 27 2016Sep 09 2016Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and ... More

More notions of forcing add a Souslin treeJul 24 2016An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved ... More

On the structure of formal balls of the balanced quasi-metric domain of wordsJul 18 2016In "Denotational semantics for programming languages, balanced quasi-metrics and fixed points" (International Journal of Computer Mathematics 85 (2008), 623-630), J. Rodr\'{i}guez-L\'{o}pez, S. Romaguera and O. Valero introduced and studied a balanced ... More

On the consistency of local and global versions of Chang's ConjectureJul 17 2016Jul 25 2016We show that for many pairs of infinite cardinals $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge ... More

Definable versions of Menger's conjectureJul 16 2016Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces are /sigma-compact, ... More

The Approachability Ideal Without a Maximal SetJul 16 2016Jul 19 2016We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial ... More

Hereditarily normal manifolds of dimension > 1 may all be metrizableJul 16 2016P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercompact cardinal, and, in addition, that the manifold is hereditarily collectionwise ... More

On the definability of Menger spaces which are not /sigma-compactJul 16 2016Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space ... More

PFA(S)[S] for the massesJul 15 2016We present S. Todorcevic's method of forcing with a coherent Souslin tree over restricted iteration axioms as a black box usable by those who wish to avoid its complexities but still access its power.

PFA(S)[S] and countably compact spacesJul 15 2016We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of "every locally compact, perfectly normal space is paracompact".

Normality versus paracompactness in locally compact spacesJul 15 2016This 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 ... More

In Cohen generic extension, every countable OD set of reals belongs to the ground modelJul 11 2016It is true in the Cohen generic extension of L, the constructible universe, that every countable ordinal-definable set of reals belongs to L.

An Easton-like theorem for Zermelo-Fraenkel Set Theory without ChoiceJul 01 2016We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $\theta(\kappa)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast ... More

A new characterisation of groups amongst monoidsJun 08 2016We prove that a monoid $M$ is a group if and only if, in the category of monoids, all points over $M$ are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden which characterises groups amongst monoids as the protomodular ... More

$\mathfrak G$-bases in free (locally convex) topological vector spacesJun 06 2016Jun 26 2016We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G$-base. A topological space $X$ has a local $\mathfrak G$-base if every point $x$ of $X$ has a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ ... More

Rigid idealsMay 31 2016An ideal $I$ on a cardinal $\kappa$ is called \emph{rigid} if all automorphisms of $P(\kappa)/I$ are trivial. An ideal is called \emph{$\mu$-minimal} if whenever $G\subseteq P(\kappa)/I$ is generic and $X\in P(\mu)^{V[G]}\setminus V$, it follows that ... More

Ueda's peak set theorem for general von Neumann algebrasMay 28 2016We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to sigma-finite von Neumann algebras (that is, von Neumann algebras with a faithful state; which includes those on a separable Hilbert space, or with ... More

Universal Graphs at $\aleph_{ω_1+1}$ and Set-theoretic GeologyMay 27 2016This thesis consists of two parts: the construction of a jointly universal family of graphs, and then an exploration of set-theoretic geology. Firstly we shall construct a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ ... More

Generalized metric properties of spheres and renorming of normed spacesMay 26 2016We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove ... More

Aronszajn trees, square principles, and stationary reflectionMay 18 2016Jun 04 2016We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(\kappa)$ introduced by Brodsky and Rinot for the purpose of constructing $\kappa$-Souslin trees. Answering ... More

Towers in filters, cardinal invariants, and Luzin type familiesMay 16 2016We investigate which filters on $\omega$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[\omega]^\omega$). We prove the following results: - Many classical examples of nice tall filters contain no ... More

Is Leibnizian calculus embeddable in first order logic?May 11 2016To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on procedural questions. This would enable an account of Leibnizian procedures ... More

Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility ModelMay 10 2016We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. ... More

Universal graphs at $\aleph_{ω_1+1}$May 02 2016Starting from a supercompact cardinal we build a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$. The same technique ... More

Interpreting the infinitesimal mathematics of Leibniz and EulerMay 02 2016We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the 17th and 18th century, in the work ... More

Weighted $βγ$-summability of fuzzy functions of order $θ$Apr 15 2016The concept of weighted $\beta\gamma$ - summability of order $\theta$ in case of fuzzy functions is introduced and classified into ordinary and absolute sense. Several inclusion relations among the sets are investigated. Also we have found some suitable ... More

Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their dualsApr 14 2016We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even ... More

Weak stability of the plasma-vacuum interface problemApr 13 2016May 14 2016We consider the free boundary problem for the two-dimensional plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider ... More

Degenerate Kalman filter error covariances and their convergence onto the unstable subspaceApr 09 2016Oct 02 2016The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman filters. In particular, as emphasized in the seminal work of Anna Trevisan and co-authors, the error covariance matrix is asymptotically supported by the unstable-neutral ... More

Gruff UltrafiltersApr 08 2016Aug 12 2016We investigate the question of whether $\mathbb Q$ carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in ... More

Diagonal supercompact Radin forcingApr 06 2016Motivated by the goal of constructing a model in which there are no $\kappa$-Aronszajn trees for any regular $\kappa>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.

Generalizing random real forcing for inaccessible cardinalsMar 28 2016The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for aleph_0^aleph_0; in spite of this similarity, the Cohen forcing and Random Real Forcing have very ... More

Weak diamond and Galvin's propertyMar 22 2016We prove that the Devlin-Shelah weak diamond implies Galvin's property. On the other hand, Galvin's property is consistent with the negation of the weak diamond, and even with Martin's axiom. We show that the proper forcing axiom implies a relative to ... More

Simultaneous stationary reflection and square sequencesMar 17 2016We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

Destructibility of the tree property at $\aleph_{ω+1}$Mar 17 2016We construct a model in which the tree property holds in $\aleph_{\omega + 1}$ and it is destructible under $\text{Col}(\omega, \omega_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings. ... More

Definable maximal cofinitary groupsMar 09 2016Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a ... More

A co-analytic Cohen indestructible maximal cofinitary groupMar 07 2016Assuming that every set is constructible, we find a $\Pi^1_1$ maximal cofinitary group of permutations of $\mathbb N$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. ... More

Spatial logic of modal mu-calculus and tangled closure operatorsMar 05 2016There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus ... More

Banach spaces from a construction schemeFeb 04 2016Sep 07 2016We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable $\varepsilon$-biorthogonal system but no uncountable $\tau$-biorthogonal system for $\tau<\varepsilon$. In particular the space have no uncountable biorthogonal system. We also construct ... More

Trees and gaps from a construction schemeFeb 04 2016Aug 15 2016We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(\omega_1, \omega_1)$-gaps. As it is well known $(\omega_1, \omega_1)$-gaps can sometimes be filled in $\omega_1$-preserving ... More

The strong tree property and weak squareJan 28 2016We show that it is consistent, relative to $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$.

The strong tree property and weak squareJan 28 2016Nov 06 2016We show that it is consistent, relative to $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$.

Spherical rectanglesJan 15 2016We study spherical quadrilaterals whose angles are odd multiples of pi/2, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many ... More

Compactifications of $ω$ and the Banach space $c_0$Jan 14 2016Jan 23 2016We investigate for which compactifications $\gamma\omega$ of the discrete space of natural numbers $\omega$, the natural copy of the Banach space $c_0$ is complemented in $C(\gamma\omega)$. We show, in particular, that the separability of the remainder ... More

Data dependence of approximate current-vortex sheets near the onset of instabilityJan 14 2016The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity ... More

Existence of approximate current-vortex sheets near the onset of instabilityJan 13 2016The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity ... More

A Microscopic approach to Souslin-tree constructions. Part IJan 08 2016We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving ... More

Unsteady flows of heat-conducting non-Newtonian fluids in~Musielak-Orlicz spacesNov 16 2015Our purpose is to show the existence of weak solutions for unsteady flow of non-Newtonian incompressible nonhomogeneous, heat-conducting fluids with generalised form of the stress tensor without any restriction on its upper growth. Motivated by fluids ... More

Inclusion Matrices and the MDS ConjectureNov 11 2015Nov 06 2016Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of at least k vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= ... More

Inclusion Matrices and the MDS ConjectureNov 11 2015Nov 16 2015Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= q, then an ... More

Instantaneous Modelling and Reverse Engineering of DataConsistent Prime Models in Seconds!Nov 11 2015A theoretical framework that supports automated construction of dynamic prime models purely from experimental time series data has been invented and developed, which can automatically generate (construct) data-driven models of any time series data in ... More

Cardinal characteristics at κ in a small u(κ) modelNov 09 2015We provide a model where u(\kappa) < 2^{\kappa} for a supercompact cardinal \kappa. Garti and Shelah have provided a sketch of how to obtain such a model by modifying the construction in a paper of Dzamonja and Shelah; we provide here a complete proof ... More

Approximate current-vortex sheets near the onset of instabilityNov 03 2015The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity ... More

Definable maximal discrete sets in forcing extensionsOct 29 2015Let $\mathcal R$ be a $\Sigma^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $\Delta^1_2$ maximal ... More

Compact convex sets that admit a lower semicontinuous strictly convex functionOct 27 2015We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak$^*$ ... More

Measure contraction properties of Carnot groupsOct 20 2015May 20 2016We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group ... More

Covering properties and square principlesOct 19 2015May 04 2016Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles, $\mathrm{CP}$ and ... More

Indestructible Guessing Models and the ContinuumOct 18 2015We introduce a stronger version of an $\omega_1$-guessing model, which we call an indestructibly $\omega_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $\omega_1$-guessing models. This principle, which follows ... More

Robust reflection principlesOct 16 2015A cardinal $\lambda$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < \lambda$, $\lambda$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals ... More

Squares and narrow systemsOct 14 2015A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal $\kappa$ satisfies the \emph{narrow system property} ... More

A general tool for consistency results related to I1Oct 12 2015In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing notions involved. ... More

CZF and Second Order ArithmeticOct 02 2015CZF + Separation is shown to be equiconsistent with second-order arithmetic, using realizability.

Independence Results around Constructive ZFOct 02 2015Using Kripke models, it is shown that CZF does not prove Power Set, and that CZF with Subset Collection substituted by Exponentiation does not prove Subset Collection.

IKP and FriendsOct 02 2015The basics of Intuitionistic Kripke-Platek set theory are developed, and some independence results among related classically equivalent theories are shown using Kripke models.

About the review in Mathematical Reviews of my paper: The two-cardinal problem for languages of arbitrary cardinality The Journal of Symbolic Logic 75, Number 3, Sept., 2010, pp. 785-801Sep 28 2015In this note I make some comments on the review MR2723767 (2011m:03064) appeared in Mathematical Reviews

$2^{\aleph_0}$ pairwise non-isomorphic maximal-closed subgroups of Sym$(\mathbb{N})$ via the classification of the reducts of the Henson digraphsSep 25 2015Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts ... More

$μ$-Abstract Elementary Classes and other generalizationsSep 24 2015Feb 11 2016We introduce $\mu$-Abstract Elementary Classes ($\mu$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $\mu$-AECs correspond ... More

On homogeneous ultrametric spacesSep 14 2015A metric space M is homogeneous if every isometry between finite subsets extends to a surjective isometry defined on the whole space. We show that if M is an ultrametric space, it suffices that isometries defined on singletons extend, i.e that the group ... More

On homogeneous ultrametric spacesSep 14 2015Nov 29 2016A metric space M is homogeneous if every isometry between finite subsets extends to a surjective isometry defined on the whole space. We show that if M is an ultrametric space, it suffices that isometries defined on singletons extend, i.e that the group ... More

Far points and discretely generated spacesSep 04 2015Feb 10 2016We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the ... More