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Strongly Luzin set Absoluteness Cardinal function Prikry-type forcing Weakly compact cardinal free Souslin tree OCA Local Club Condensation. S-Space Chang's conjecture transformations square principles positive partition relation SNR strongly bounded groups Forcing Axioms L-space Distributive tree Closed coloring Partition relations for trees very good scale full tree Postprocessing function Parameterized proxy principle Almost-disjoint family Slim tree Rado's conjecture Knaster Intersection model Successor of Singular Cardinal ccc Fat stationary set projective Boolean algebra xbox club_AD Knaster and friends Respecting tree perfectly normal Uniformly homogeneous Luzin set stationary reflection Well-behaved magma Nonspecial tree countably metacompact Greatly Mahlo Club Guessing Poset incompactness Almost Souslin O-space Aronszajn tree Analytic sets P-Ideal Dichotomy Subnormal ideal Monotonically far b-scale free Boolean algebra Commutative cancellative semigroups reflection principles diamond star Minimal Walks 54G20 Square-Brackets Partition Relations Partition Relations nonmeager set Souslin Tree Lipschitz reduction Open Access Forcing Jonsson cardinal unbounded function HOD Amenable C-sequence Strong coloring Erdos Cardinal Non-saturation Almost countably chromatic ZFC construction Ramsey theory over partitions AIM forcing Mandelbrot set Strongly compact cardinal Ascent Path Chromatic number Cohen real Sakurai's Bell inequality Uniformly coherent Diamond Generalized Clubs Forcing with side conditions Microscopic Approach Selective Ultrafilter Fodor-type reflection Successor of Regular Cardinal Prevalent singular cardinals C-sequence weak diamond PFA specializable Souslin tree Singular cofinality Shelah's Strong Hypothesis Subadditive Singular cardinals combinatorics Was Ulam right? Whitehead Problem Kurepa Hypothesis Large Cardinals Filter reflection Dowker space Vanishing levels Small forcing sap Uniformization Reduced Power Antichain Precaliber approachability ideal Foundations Generalized descriptive set theory Hereditarily Lindelöf space middle diamond Universal Sequences Martin's Axiom Diamond-sharp Sigma-Prikry Ineffable cardinal square Diamond for trees Axiom R Dushnik-Miller Erdos-Hajnal graphs Commutative projection system GMA Coherent tree tensor product graph weak Kurepa tree Rainbow sets Cardinal Invariants Ostaszewski square Subtle cardinal Entangled linear order Interval topology on trees higher Baire space Sierpinski's onto mapping principle Countryman line stationary hitting Subtle tree property weak square super-Souslin tree indecomposable filter Reflecting stationary set Ascending path Rock n' Roll Ulam matrix polarized partition relation Iterated forcing stick PFA(S)[S] coloring number Fast club Constructible Universe regressive Souslin tree Hindman's Theorem Singular Density Hedetniemi's conjecture
Tag Archives: Successor of Singular Cardinal
Perspectives on Set Theory, November 2023
I gave an invited talk at the Perspectives on Set Theory conference, November 2023. Talk Title: May the successor of a singular cardinal be Jónsson? Abstract: We’ll survey what’s known about the question in the title and collect ten open … Continue reading
Posted in Invited Talks, Open Problems
Tagged Jonsson cardinal, Successor of Singular Cardinal
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Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. 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 classical results of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
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A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading
A relative of the approachability ideal, diamond and non-saturation
Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
