An easy proof that there may be no P-points
We provide an easy proof that there are no P-points in the models obtained by adding many Silver reals with countable support to a model of CH.
math.LO↗
arXiv subjects
Osvaldo Guzmán
Publications and source records attributed to Osvaldo Guzmán.
Properties of Laver forcing associated with a co-ideal expressed via the Katetov order
We study variants of classical Laver forcing defined from co-ideals and analyze their combinatorial properties in terms of the Katětov order. In particular, we give a Katětov-theoretic characterization of when Laver forcing associated with a co-ideal adds Cohen reals, and we show that such forcings never add random reals. Improving a result of Błaszczyk and Shelah we prove that the addition of Cohen reals and the Laver property are not equivalent, even in the case of ultrafilters. As an application, we investigate the problem of adding half Cohen reals without adding Cohen reals via tree-like forcings arising from co-ideals. We obtain several partial results and structural obstructions. Finally, we resolve several open questions from the literature concerning the one-to-one or constant property and cardinal invariants associated with ideals.
math.LO↗
Are scales Fréchet?
We continue the study of Dow spaces of a $\mathfrak{b}$-scale, originally introduced by Alan Dow in "$π$-Weight and the Fréchet-Urysohn property" (Topology and its Applications, Vol. 174, pp. 56-61). We prove that it is consistent that all such spaces are Fréchet, but it is also consistent that none of them is. We use these spaces to exhibit (consistently) a $\triangle_{2}^{1}$ ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.
math.GN↗
Diamonds on trees
We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $ω_1$-tree, then $\diamondsuit_T \implies \diamondsuit$, and if $T$ is a Suslin tree, then $\diamondsuit_T \iff \diamondsuit$. We also prove that $\diamondsuit^*$ implies $\diamondsuit_T$ (yielding the consistency of $\diamondsuit_T$) and establish the consistency of $\neg\diamondsuit^* + (\forall T\text{ nonspecial }ω_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $ω_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $σ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations.
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Shelah Ultrafilters
In this paper, we study a special type of ultrafilter which we call Shelah ultrafilter. We show that it is possible to add a Shelah ultrafilter using a special forcing notion. We also show that Shelah ultrafilters turn out to be I-ultrafilters for many Borel ideals.
math.LO↗
No covering with nowhere dense \textsf{P}-sets in the Cohen model
We prove that if less than $\aleph_ω$-many Cohen reals are added to a model of \textsf{CH}, then $ω^{\ast}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, there is an ultrafilter on $ω$ that does not contain a tower).
math.LO↗
Ultrafilters in the random real model
We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph_ω% $-many random reals to a model of \textsf{CH. }These results improve and correct previous theorems that can be found in the literature.
math.LO↗
The Category Dichotomy for Ideals
We provide a counterexample to the Category Dichotomy in the framework of $\textsf{ZFC}$. That is, we prove the existence of an ideal on $ω$ that is not Katětov below $\mathsf{nwd}$ and does not have restrictions above $\mathcal{ED}$. We also prove that in the Laver model every tall $\mathsf{P}$-ideal is Katětov-Blass above $\mathcal{ED}_{\mathsf{fin}}$ and that it is consistent that every $\mathsf{Q}^{+}$ ideal is meager.
math.LO↗
High dimensional sequential compactness
We give examples of $n$-sequentially compact spaces that are not $(n+1)$-sequentially compact under several assumptions. We improve results from Kubis and Szeptycki by building such examples from $\mathfrak{b=c}$ and $\diamondsuit(\mathfrak{b})+\mathfrak{d}=ω_1$. We also introduce a new splitting-like cardinal invariant and then show that the same holds under $\mathfrak{s=b}$.
math.GN↗
Construction schemes: transferring structures from $ω$ to $ω_1$
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of capturing axioms associated to construction schemes. From them, we deduce the existence of several uncountable structures which are known to be independent from the usual axioms of Set Theory. Lastly, we prove that the capturing axiom $FCA(part)$ is implied by Jensen's $\Diamond$ principle.
math.LO↗
Almost disjoint families and the geometry of nonseparable spheres
We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some new almost disjoint families among others we show the existence of Banach spaces where the unit spheres display surprising geometry: 1) There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than $1$. 2) It is consistent that for every $ρ>0$ there is a nonseparable Banach space, where for every $δ>0$ there is $\varepsilon>0$ such that every uncountable $(1-\varepsilon)$-separated set of elements of the unit sphere contains two elements distant by less than $1$ and two elements distant at least by $2-ρ-δ$. It should be noted that for every $\varepsilon>0$ every nonseparable Banach space has a plenty of uncountable $(1-\varepsilon)$-separated sets by the Riesz Lemma. We also obtain a consistent dichotomy for the spaces of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$: The Open Coloring Axiom implies that the unit sphere of every Banach space of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ either is the union of countably many sets of diameter strictly less than $1$ or it contains an uncountable $(2-\varepsilon)$-separated set for every $\varepsilon>0$.
math.FA↗
Combinatorial properties of MAD families
We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns if for every set $X\subseteq{\left[ ω\right]}^{<ω}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Steprāns if and only if it is Katětov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Steprāns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Steprāns families of size ${\aleph}_{1}$.
math.LO↗
HL ideals and Sacks indestructible ultrafilters
We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the ideal $\mathcal Z$ of sets of asymptotic density zero. We investigate the existence of Sacks indestructible ultrafilters and prove that every Sacks indestructible ultrafilter is a $\mathcal Z$-ultrafilter.
math.LO↗
Maximal almost disjoint families and pseudocompactness of hyperspaces
We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(ω)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost disjoint family $\mathcal A$ of size less than $\mathfrak c$ whose hyperspace is not pseudocompact.
math.GN↗
Splitting chains, tunnels and twisted sums
We study splitting chains in $\mathscr{P}(ω)$, that is, families of subsets of $ω$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of $C(ω^*)=\ell_\infty/c_0$ and $c_0(\mathfrak c)$. Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.
math.LO↗
On $\mathbb R$-embeddability of almost disjoint families and Akemann-Doner C*-algebras
An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost disjoint family of size continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost disjoint family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost disjoint family, where no two uncountable subfamilies can be separated but always a countable subfamily can be separated from any disjoint subfamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $ω_1$ with no nonseparable commutative subalgebra.
math.LO↗
There are no P-points in Silver extensions
We prove that after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this remains to be the case in any further extension which has the Sacks property. We conclude that there are no P-points in the Silver model. In particular, it is possible to construct a model without P-points by iterating Borel partial orders. This answers a question of Michael Hrušák. We also show that the same argument can be used for the side-by-side product of Silver forcing. This provides a model without P-points with the continuum arbitrary large, answering a question of Wolfgang Wohofsky.
math.LO↗
Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters
We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $ω$-hitting and $ω$-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.
math.LO↗