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GMA Dowker space Club Guessing Axiom R Chromatic number unbounded function AIM forcing Parameterized proxy principle O-space Diamond-sharp Commutative cancellative semigroups weak diamond Postprocessing function Successor of Singular Cardinal Almost Souslin Prikry-type forcing Forcing Axioms Souslin Tree Forcing with side conditions approachability ideal Uniformly homogeneous Commutative projection system stick Reduced Power Fat stationary set club_AD Fast club perfectly normal Generalized Clubs Singular Density Subadditive PFA Subtle cardinal Hindman's Theorem higher Baire space Partition relations for trees Interval topology on trees reflection principles Was Ulam right? Lipschitz reduction Closed coloring Slim tree Local Club Condensation. Absoluteness S-Space Minimal Walks Ascent Path Strong coloring Entangled linear order Greatly Mahlo Amenable C-sequence Uniformization specializable Souslin tree Well-behaved magma Shelah's Strong Hypothesis Rainbow sets Subnormal ideal Aronszajn tree Luzin set Whitehead Problem Nonspecial tree Large Cardinals stationary reflection Small forcing Strongly compact cardinal ccc Universal Sequences Singular cofinality Chang's conjecture middle diamond incompactness ZFC construction Microscopic Approach Successor of Regular Cardinal Countryman line Sakurai's Bell inequality Subtle tree property L-space Ostaszewski square Mandelbrot set Erdos Cardinal Erdos-Hajnal graphs Open Access Almost countably chromatic transformations Cardinal Invariants Coherent tree Antichain indecomposable filter Jonsson cardinal Fodor-type reflection Poset Respecting tree P-Ideal Dichotomy Partition Relations sap Knaster Hereditarily Lindelöf space Diamond for trees b-scale Monotonically far Dushnik-Miller OCA full tree Generalized descriptive set theory strongly bounded groups C-sequence Selective Ultrafilter Sierpinski's onto mapping principle PFA(S)[S] 54G20 Cardinal function Non-saturation Reflecting stationary set countably metacompact Knaster and friends Ascending path weak Kurepa tree Diamond regressive Souslin tree stationary hitting Distributive tree nonmeager set coloring number Intersection model Singular cardinals combinatorics xbox Rado's conjecture super-Souslin tree Almost-disjoint family Hedetniemi's conjecture positive partition relation projective Boolean algebra weak square square Sigma-Prikry Prevalent singular cardinals Cohen real Martin's Axiom Square-Brackets Partition Relations Rock n' Roll square principles Analytic sets HOD tensor product graph Iterated forcing SNR Precaliber Foundations very good scale Vanishing levels polarized partition relation Ineffable cardinal diamond star Kurepa Hypothesis Weakly compact cardinal Constructible Universe Uniformly coherent Filter reflection free Souslin tree Ramsey theory over partitions Forcing free Boolean algebra Ulam matrix Strongly Luzin set
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
