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- The S-space problem, and the cardinal invariant $\mathfrak b$ April 4, 2013
- An $S$-space from a Cohen real April 3, 2013
- Forcing with a Souslin tree makes $\mathfrak p=\omega_1$ April 1, 2013
- The S-space problem, and the cardinal invariant $\mathfrak p$ March 28, 2013
- Jones’ theorem on the cardinal invariant $\mathfrak p$ March 26, 2013
- Erdős 100 March 26, 2013
- Bell’s theorem on the cardinal invariant $\mathfrak p$ March 21, 2013
- The $\Delta$-system lemma: an elementary proof March 20, 2013
Keywords
Successor of Regular Cardinal Club Guessing polarized partition relation stationary reflection Chromatic number very good scale Shelah's Strong Hypothesis Forcing free Boolean algebra Partition Relations Antichain Singular cardinals combinatorics Dushnik-Miller Cohen real Minimal Walks Souslin Tree Diamond Poset Whitehead Problem Erdos-Hajnal graphs PFA(S)[S] Knaster Small forcing Singular Density Cardinal function approachability ideal S-Space weak diamond Rainbow sets sap Rado's conjecture Prikry-type forcing Prevalent singular cardinals P-Ideal Dichotomy Ostaszewski square Rock n' Roll Generalized Clubs Foundations Non-saturation Almost countably chromatic Aronszajn tree Singular Cofinality reflection principles Sakurai's Bell inequality Mandelbrot set Erdos Cardinal Uniformization b-scale Large Cardinals projective Boolean algebra Successor of Singular Cardinal Kurepa Hypothesis Axiom R diamond star weak square stationary hitting square incompactness Hereditarily Lindelöf space middle diamond Square-Brackets Partition Relations
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
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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
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
