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

# 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