Regular cardinal
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On -Strongly Measurable Cardinals Open
We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o…
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Δ<sub>1</sub>-Definability of the non-stationary ideal at successor cardinals Open
Assuming $V=L$, for every successor cardinal $\kappa $ we construct a GCH and cardinal preserving forcing poset $\mathbb {P}\in L$ such that in $L^{\mathbb {P}}$ the ideal of all non-stationary subsets of $\kappa $ is $\Delta _1$-definable…
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INDESTRUCTIBILITY OF THE TREE PROPERTY Open
In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$ , and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at…
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Blurry Definability Open
I begin the study of a hierarchy of (hereditarily) <κ-blurrily ordinal definable sets. Here for a cardinal κ, a set is <κ-blurrily ordinal definable if it belongs to an OD set of cardinality less than κ, and it is hereditarily so if it and…
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ON RESTRICTIONS OF ULTRAFILTERS FROM GENERIC EXTENSIONS TO GROUND MODELS Open
Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter 1 , 2 W over $\kappa $ . Set $U=W\cap V$ .
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Controlling classical cardinal characteristics while collapsing cardinals Open
We show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and the values in Cichoń's diagram, using the Boolean Ultrapower method. In our recent paper [J. Math. Logic 21 (2021)] the same was done for a newer Ci…
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Definable orthogonality classes in accessible categories are small Open
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka's principle. We prove that the necessary large-cardinal hypoth…
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On Borel reducibility in generalized Baire space Open
We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $\kappa ^\kappa $ for an uncountable $\kappa $ with $\kappa ^{<\kappa }=\kappa $. The theory looks quite different from its classical counterpart…
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Transformations of the transfinite plane Open
We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we pr…
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MORE ON THE PRESERVATION OF LARGE CARDINALS UNDER CLASS FORCING Open
We prove two general results about the preservation of extendible and $C^{(n)}$ -extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Prin…
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ON GENERALISED METRISABILITY AND CARDINAL INVARIANTS IN QUASITOPOLOGICAL GROUPS Open
We consider generalised metrisability and cardinal invariants in quasitopological groups. We construct examples to show that some equalities of cardinal invariants in topological groups cannot be extended to quasitopological groups.
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Galvin's property at large cardinals and an application to partition calculus Open
In the first part of this paper, we explore the possibility for a very large cardinal $κ$ to carry a $κ$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $κ$-complete ultrafilt…
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The Seed Order Open
This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even un…
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The higher Cichoń diagram in the degenerate case Open
For a regular uncountable cardinal $\kappa$, we discuss the order relationship between the unbounding and dominating numbers $\mathfrak{b}_{\kappa}$ and $\mathfrak{d}_{\kappa}$ on $\kappa$ and cardinal invariants of the higher meager ideal…
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Anti-Urysohn spaces Open
All spaces are assumed to be infinite Hausdorff spaces. We call a space anti-Urysohn (AU in short) iff any two non-empty regular closed sets in it intersect. We prove that for every infinite cardinal κ there is a space of size κ in which f…
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On non-forking spectra Open
Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order…
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Contributions to the Theory of Large Cardinals through the Method of Forcing Open
The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms. The dissertation is divided into two thematic blocks. In Bloc…
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The Halpern-Läuchli Theorem at a Measurable Cardinal Open
Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For $κ$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on a…
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Two inequalities between cardinal invariants Open
We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an …
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-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders Open
Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the …
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Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions Open
Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal $\\kappa$ . We show the consistency of $…
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Towers and gaps at uncountable cardinals Open
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $…
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Generalized Choquet spaces Open
We introduce an analog to the notion of Polish space for spaces of weight\n$\\leq\\kappa$, where $\\kappa$ is an uncountable regular cardinal such that\n$\\kappa^{<\\kappa}=\\kappa$. Specifically, we consider spaces in which player II\nhas…
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Strong Chang’s Conjecture, Semi-Stationary Reflection, the Strong Tree Property and two-cardinal square principles Open
We prove that a strong version of Chang’s Conjecture implies both the Strong Tree Property for $\omega _2$ and the negation of the square principle $\square (\lambda , \omega )$ for every regular cardinal $\lambda \geq \omega _2$.
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DENSE IDEALS AND CARDINAL ARITHMETIC Open
From large cardinals we show the consistency of normal, fine, κ -complete λ -dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ . We explore the interplay between dense ideals, cardinal arithmetic, and squares, answ…
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A Galvin–Hajnal theorem for generalized cardinal characteristics Open
We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin–Hajnal theorem, and hence also of Silver’s theorem, at singular ca…
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A Forcing Axiom Deciding the Generalized Souslin Hypothesis Open
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\unicode[STIX]{x1D706}$ , …
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Two Upper Bounds on Consistency Strength of ¬□ℵω and Stationary Set Reflection at Two Successive ℵn Open
We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a $\\kappa^{+}$ -supercompact cardinal. All previous…
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Ordinal compactness Open
We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities. The most general form depends on two ordinal parameters. Ordinal compactness turns out to be a much mor…
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THE WEAK VOPĚNKA PRINCIPLE FOR DEFINABLE CLASSES OF STRUCTURES Open
We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each…