Archives
Keywords
projective Boolean algebra Shelah's Strong Hypothesis Fodor-type reflection full tree Well-behaved magma indecomposable filter weak Kurepa tree Amenable C-sequence stationary reflection Ineffable cardinal Successor of Regular Cardinal Generalized descriptive set theory Foundations Almost-disjoint family Prevalent singular cardinals Monotonically far reflection principles Hindman's Theorem Ostaszewski square Subtle tree property Hereditarily Lindelöf space Cardinal function Rainbow sets Filter reflection Souslin Tree super-Souslin tree Kurepa Hypothesis PFA Ramsey theory over partitions Iterated forcing approachability ideal PFA(S)[S] weak square Singular Density Antichain nonmeager set unbounded function Prikry-type forcing Fast club Lipschitz reduction Postprocessing function Minimal Walks Dushnik-Miller Strongly Luzin set polarized partition relation square principles Luzin set AIM forcing square Respecting tree Intersection model Commutative projection system Rado's conjecture Absoluteness Precaliber Martin's Axiom Diamond for trees stationary hitting free Souslin tree Dowker space ccc Ascent Path very good scale Knaster and friends Partition Relations Uniformization Universal Sequences Ascending path Almost countably chromatic Entangled linear order Forcing with side conditions Erdos-Hajnal graphs Small forcing SNR Chromatic number 54G20 b-scale C-sequence Axiom R Erdos Cardinal L-space Whitehead Problem Interval topology on trees Slim tree Subnormal ideal Knaster GMA Chang's conjecture Non-saturation O-space Forcing Axioms HOD Reduced Power Cardinal Invariants Forcing Sigma-Prikry Diamond-sharp Strong coloring xbox countably metacompact Reflecting stationary set Subadditive Almost Souslin Mandelbrot set OCA tensor product graph Subtle cardinal Successor of Singular Cardinal sap regressive Souslin tree Microscopic Approach Partition relations for trees Uniformly coherent Poset Countryman line Closed coloring Selective Ultrafilter Nonspecial tree S-Space Generalized Clubs perfectly normal Analytic sets Strongly compact cardinal Commutative cancellative semigroups Rock n' Roll P-Ideal Dichotomy Aronszajn tree Club Guessing transformations Sakurai's Bell inequality positive partition relation Ulam matrix Cohen real Parameterized proxy principle Hedetniemi's conjecture higher Baire space Diamond specializable Souslin tree Uniformly homogeneous Distributive tree Coherent tree Open Access Local Club Condensation. weak diamond coloring number Large Cardinals Vanishing levels ZFC construction Weakly compact cardinal Sierpinski's onto mapping principle Fat stationary set free Boolean algebra Constructible Universe incompactness Jonsson cardinal middle diamond diamond star Singular cardinals combinatorics Greatly Mahlo stick club_AD Singular cofinality strongly bounded groups Square-Brackets Partition Relations Was Ulam right?
Tag Archives: approachability ideal
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
Leave a comment
Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … 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
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
