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