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L-space projective Boolean algebra Forcing Axioms Knaster Cohen real Minimal Walks Foundations Diamond square Prikry-type forcing Weakly compact cardinal Axiom R very good scale Chromatic number diamond star Cardinal function Erdos-Hajnal graphs Successor of Regular Cardinal Aronszajn tree Large Cardinals polarized partition relation Ostaszewski square Prevalent singular cardinals Mandelbrot set Rado's conjecture Antichain Cardinal Invariants Square-Brackets Partition Relations Rock n' Roll P-Ideal Dichotomy PFA Forcing Erdos Cardinal Partition Relations Hereditarily Lindelöf space PFA(S)[S] Hedetniemi's conjecture Uniformization Sakurai's Bell inequality Singular Cofinality Universal Sequences Small forcing Constructible Universe weak square Successor of Singular Cardinal Shelah's Strong Hypothesis weak diamond ccc b-scale S-Space Kurepa Hypothesis Almost countably chromatic Absoluteness Whitehead Problem tensor product graph Martin's Axiom OCA Generalized Clubs Almost-disjoint famiy Dushnik-Miller stationary hitting Singular cardinals combinatorics Poset incompactness reflection principles Club Guessing sap middle diamond stationary reflection Singular Density Non-saturation Souslin Tree free Boolean algebra Rainbow sets approachability ideal

# Tag Archives: approachability ideal

## 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
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## 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

Posted in Open Problems, Publications
Tagged 03E05, 03E35, 03E50, approachability ideal, Club Guessing, Diamond, diamond star, Non-saturation, sap, Souslin Tree, square, stationary hitting, Uniformization, Whitehead Problem
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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