arXiv subjects

Alejandro Poveda

Publications and source records attributed to Alejandro Poveda.

At least 19 recordsLinked to original sources

arXiv2026

Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem

We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg--Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_\omega$ can be the first place where $\mathcal{P}(\aleph_\omega)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_\omega)$, answering a question of Hayut. (3) If $\kappa$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $\kappa^+$ and the GCH holds in $\mathrm{HOD}$ below $\kappa^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(\kappa)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.

math.LO↗

arXiv2026

The number of measures on very large measurable cardinals

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first $n$ measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.

math.LO↗

arXiv2026

The directedness of the Rudin-Keisler order at measurable cardinals

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC: (1) $\kappa$ is a measurable cardinal with $o(\kappa)=\lambda^+$ (resp. $o(\kappa)=\kappa$). (2) The Rudin-Keisler order restricted to the set of $\kappa$-complete (non-principal) ultrafilters on $\kappa$ is $\lambda^+$-directed (resp. $\kappa^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\lambda$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed.

math.LO↗

arXiv2025

Combinatorics in Higher Solovay Models

We construe the singular-cardinal analogue of the classical Solovay model. Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\mathcal{P}(\kappa))$ that are typically implied by Woodin's axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $\kappa^+$-sequences of distinct members of $\mathcal{P}(\kappa)$ and that Shelah's approachability property $\AP_\kappa$ fails. We prove that every set in our inner model satisfies a singular analogue of the complete Ramsey property and that the partition relation $\kappa\xrightarrow[]{\mathrm{OD}} (\omega)^\omega_{V_\mu}$ holds for all $\mu<\kappa$.

math.LO↗

arXiv2025

On Shelah's Approachability Ideal

We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[\kappa^+]$. Given a singular cardinal $\aleph_\gamma$, a regular cardinal $\mu\in (\mathrm{cf}(\gamma),\aleph_\gamma)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which $\aleph_{\gamma+1} \cap \mathrm{cof}(\mu) \notin I[\aleph_{\gamma+1}]$. This provides a definitive answer to a question of Shelah from the 80's. In addition, assuming large cardinals, we construct a model of $\mathsf{ZFC}$ in which the approachability property fails, simultaneously, at every singular cardinal. This is a major milestone in the solution of a question of Foreman and Magidor from the 80's.

math.LO↗

arXiv2025

Isomorphism Classes of Generating Sets

We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $\sigma$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $\omega$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $\kappa$ is supercompact, and for a $\kappa^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}\kappa$-directed closed $\kappa^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $\kappa$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_\kappa<2^\kappa$, answering questions from \cite{Benhamou_Goldberg2025}.

math.LO↗

arXiv2025

A Banach space with $L$-orthogonal sequences but without $L$-orthogonal elements

We prove that the existence of Banach spaces with $L$-orthogonal sequences but without $L$-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to \cite[Question~1.1]{AvilesMartinezRueda}. Generalizing classical $Q$-point ultrafilters, we introduce the notion of $Q$-measures and provide several results generalizing former theorems by Miller \cite{Miller} and Bartoszynski \cite{Bartoszynski} for $Q$-point ultrafilters.

math.LO↗

arXiv2024

On the optimality of the HOD dichotomy

In the first part of the manuscript, we establish several consistency results concerning Woodin's $\HOD$ hypothesis and large cardinals around the level of extendibility. First, we prove that the first extendible cardinal can be the first strongly compact in HOD. We extend a former result of Woodin by showing that under the HOD hypothesis the first extendible cardinal is $C^{(1)}$-supercompact in HOD. We also show that the first cardinal-correct extendible may not be extendible, thus answering a question by Gitman and Osinski \cite[\S9]{GitOsi}. In the second part of the manuscript, we discuss the extent to which weak covering can fail below the first supercompact cardinal $\delta$ in a context where the HOD hypothesis holds. Answering a question of Cummings et al. \cite{CumFriGol}, we show that under the $\HOD$ hypothesis there are many singulars $\kappa<\delta$ where $\cf^{\HOD}(\kappa)=\cf(\kappa)$ and $\kappa^{+\HOD}=\kappa^{+}.$ In contrast, we also show that the $\HOD$ hypothesis is consistent with $\delta$ carrying a club of $\HOD$-regulars cardinals $\kappa$ such that $\kappa^{+\HOD}<\kappa^{+}$. Finally, we close the manuscript with a discussion about the $\HOD$ hypothesis and $\omega$-strong measurability.

math.LO↗

arXiv2024

The Baire and perfect set properties at singulars cardinals

We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa+1})$ has both the $\kappa$-Perfect Set Property and the $\mathcal{\vec{U}}$-Baire Property. This is a higher analogue of Solovay's result for $L(\mathbb{R})$. We obtain this configuration starting with large-cardinal assumptions in the realm of supercompactness, thus improving former theorems by Cramer, Shi and Woodin.

math.LO↗

arXiv2024

Almost free modules, perfect decomposition and Enochs's conjecture

Given a module $X$ and a regular cardinal $\kappa$ we study various notions of $(\kappa,\mathrm{Add}(X))$-freeness and $(\kappa,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $\kappa^+$-generated, $(\kappa^+,\mathrm{Add}(X))$-free and $(\kappa^+,\mathrm{Add}(X))$-separable module. Our construction allows $\kappa$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by \v{S}aroch \cite{Saroch}.

math.RA↗

arXiv2024

Axiom $\mathcal{A}$ and supercompactness

We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to a question of Bagaria \cite[p.19]{Bag}. This configuration is a consequence of a new axiom we introduce -- called $\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.

math.LO↗

arXiv2024

Approximation properties of torsion classes

We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in \cite{Cox_MaxDecon} requires large cardinals; it sits, implication-wise, between Vop\v{e}nka's Principle and the existence of an $\omega_1$-strongly compact cardinal. (2) While deconstructibility of a class of modules always implies the precovering property by \cite{MR2822215}, the concepts are (consistently) non-equivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

math.LO↗

arXiv2024

Non-Normal Magidor-Radin Types of Forcings

We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}. Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $\omega$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $\kappa$ to $\omega_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $\kappa$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $\kappa$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $\kappa$ to $\omega_1$ while introducing a Cohen subset of $\kappa$.

math.LO↗

arXiv2023

On the property (C) of Corson and other sequential properties of Banach Spaces

A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a countable subset of $C$. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of $B_{X^*}$ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under $\mathrm{MA}+\neg \mathrm{CH}$ for the class of $\mathcal{C}(K)$-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom ($\mathrm{PFA}$) the following conditions are equivalent for an arbitrary Banach space $X$: 1) $X$ has property $\mathcal{E}'$; 2) $X$ has weak*-sequential dual ball; 3) $X$ has property ($\mathcal{C}$) of Corson; 4) $(B_{X^*},w^\ast)$ has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property $\mathcal{E}'$ has weak*-convex block compact dual ball.

math.FA↗

arXiv2022

The Gluing Property

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many $\kappa$-complete ultrafilters on $\kappa$ can be glued into a $\kappa$-complete extender. We show that every $\kappa$-compact cardinal has the $2^\kappa$-gluing property, yet non-necessarily the $(2^\kappa)^+$-gluing property. Finally, we compute the exact consistency-strength for $\kappa$ to have the $\omega$-gluing property; this being $o(\kappa)=\omega_1$.

math.LO↗

arXiv2022

Non-Galvin Filters

We address the question of the consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions \cite[Questions 7.8,7.9]{TomMotiII}, \cite[Question 5]{NegGalSing} and improve theorem \cite[Theorem 2.3]{NegGalSing}.

math.LO↗

arXiv2022

Sigma-Prikry forcing III: Down to Aleph_omega

We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.

math.LO↗

arXiv2022

Galvin's property at large cardinals and an application to partition calculus

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $\kappa$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $\kappa$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. Finally, we apply this property to obtain consistently new instances of the classical problem in partition calculus $\lambda\rightarrow(\lambda,\omega+1)^2$.

math.LO↗