Archives
Keywords
Uniformization perfectly normal Rado's conjecture Well-behaved magma Sierpinski's onto mapping principle Prevalent singular cardinals Selective Ultrafilter Prikry-type forcing Axiom R Entangled linear order Strong coloring GMA Generalized Clubs Shelah's Strong Hypothesis Ulam matrix PFA(S)[S] Dowker space regressive Souslin tree Distributive tree Forcing with side conditions Coherent tree club_AD stationary hitting Local Club Condensation. Forcing Axioms Subtle tree property ccc Intersection model Precaliber Foundations Respecting tree Cardinal function Minimal Walks Almost countably chromatic Knaster and friends Erdos-Hajnal graphs nonmeager set Mandelbrot set Cohen real super-Souslin tree xbox Almost Souslin Subtle cardinal Microscopic Approach Vanishing levels Open Access incompactness approachability ideal Forcing indecomposable filter projective Boolean algebra Monotonically far Kurepa Hypothesis b-scale Was Ulam right? Analytic sets very good scale Filter reflection Diamond-sharp polarized partition relation L-space sap Postprocessing function Universal Sequences Interval topology on trees Sakurai's Bell inequality positive partition relation Singular Density Jonsson cardinal Hedetniemi's conjecture Singular cardinals combinatorics Sigma-Prikry Strongly Luzin set SNR Rock n' Roll Nonspecial tree unbounded function Uniformly homogeneous Closed coloring C-sequence Fat stationary set Club Guessing Luzin set Ostaszewski square Weakly compact cardinal Dushnik-Miller Knaster Antichain weak diamond free Boolean algebra P-Ideal Dichotomy Small forcing transformations specializable Souslin tree Greatly Mahlo Iterated forcing O-space Slim tree Countryman line Martin's Axiom Successor of Singular Cardinal tensor product graph Strongly compact cardinal Partition Relations Rainbow sets Parameterized proxy principle PFA Subnormal ideal coloring number Partition relations for trees Poset Ineffable cardinal Non-saturation HOD Ramsey theory over partitions Chromatic number Chang's conjecture Hereditarily Lindelöf space Ascending path Uniformly coherent Successor of Regular Cardinal Whitehead Problem weak square countably metacompact reflection principles ZFC construction 54G20 Lipschitz reduction free Souslin tree square principles Reduced Power Fodor-type reflection AIM forcing Aronszajn tree Commutative cancellative semigroups Diamond Reflecting stationary set Subadditive higher Baire space Amenable C-sequence stick Square-Brackets Partition Relations Commutative projection system Generalized descriptive set theory Erdos Cardinal S-Space stationary reflection Cardinal Invariants full tree diamond star Hindman's Theorem middle diamond Fast club Ascent Path Absoluteness Singular cofinality square Diamond for trees strongly bounded groups Almost-disjoint family weak Kurepa tree Souslin Tree Constructible Universe OCA Large Cardinals
Category Archives: Publications
The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
1 Comment
Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … Continue reading
Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
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
Leave a comment
On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
A topological reflection principle equivalent to Shelah’s strong hypothesis
Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading
Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Open Access, Shelah's Strong Hypothesis
Comments Off on A topological reflection principle equivalent to Shelah’s strong hypothesis
Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
Comments Off on Openly generated Boolean algebras and the Fodor-type reflection principle
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
On guessing generalized clubs at the successors of regulars
Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading
