Results 21 to 30 of about 199 (115)
Iteration of λ‐complete forcing notions not collapsing λ+
International Journal of Mathematics and Mathematical Sciences, Volume 28, Issue 2, Page 63-82, 2001., 2001 We look for a parallel to the notion of “proper forcing” among λ‐complete forcing notions not collapsing λ+. We suggest such a definition and prove that it is preserved by suitable iterations.
Andrzej Rosłanowski, Saharon Shelah
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International Journal of Mathematics and Mathematical Sciences, Volume 13, Issue 3, Page 507-512, 1990., 1989 An ideal on a set X is a nonempty collection of subsets of X closed under the operations of subset (heredity) and finite unions (additivity). Given a topological space (X, τ) an ideal ℐ on X and A⊆X, ψ(A) is defined as ⋃{U ∈ τ : U − A ∈ ℐ}. A topology, denoted τ*, finer than τ is generated by the basis {U − I : U ∈ τ, I ∈ ℐ}, and a topology, denoted 〈ψ(
T. R. Hamlett, David Rose
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Generically supercompact cardinals by forcing with chain conditions
, 2022 A ccc-generically supercompact cardinal $κ$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $κ$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1).
Fuchino, Sakaé, Sakai, Hiroshi
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Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics
Bulletin of Symbolic Logic, 2008 The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing ...
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REFLECTION IN SECOND-ORDER SET THEORY WITH ABUNDANT URELEMENTS BI-INTERPRETS A SUPERCOMPACT CARDINAL
The Journal of Symbolic Logic, 2022 AbstractAfter reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove that second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence
Joel David Hamkins, Bokai Yao
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The Partition Property for Certain Extendible Measures on Supercompact Cardinals [PDF]
Proceedings of the American Mathematical Society, 1981 ABsTRAcr. We give an alternate characterization of a combinatorial property of measures onpKA introduced by Menas. We use this characterization to prove that if K is supercompact, then all measures on pKX in a certain class have the partition property. This result is applied to obtain a self-contained proof that if K iS supercompact and X is the least ...
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Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals
Journal of the London Mathematical Society, Volume 112, Issue 1, July 2025.Abstract Motivated by recent work of Boney, Dimopoulos, Gitman, and Magidor, we characterize the existence of weak compactness cardinals for all abstract logics through combinatorial properties of the class of ordinals. This analysis is then used to show that, in contrast to the existence of strong compactness cardinals, the existence of weak ...
Philipp Lücke
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Full Souslin trees at small cardinals
Journal of the London Mathematical Society, Volume 110, Issue 1, July 2024.Abstract A κ$\kappa$‐tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full κ$\kappa$‐Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal κ$\kappa $. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there
Assaf Rinot, Shira Yadai, Zhixing You
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Bulletin of the London Mathematical Society, Volume 56, Issue 6, Page 1967-1987, June 2024.Abstract We investigate connections between set‐theoretic compactness principles and cardinal arithmetic, introducing and studying generalized narrow system properties as a way to approach two open questions about two‐cardinal tree properties. The first of these questions asks whether the strong tree property at a regular cardinal κ⩾ω2$\kappa \geqslant
Chris Lambie‐Hanson
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Journal of the London Mathematical Society, Volume 109, Issue 6, June 2024.Abstract We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a strongly inaccessible cardinal κ ...
Tom Benhamou, Jing Zhang
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