### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

Absoluteness Dushnik-Miller Fat stationary set Chromatic number Sakurai's Bell inequality Selective Ultrafilter Prikry-type forcing Forcing Axioms Cardinal function HOD Forcing Rado's conjecture Commutative cancellative semigroups Almost countably chromatic reflection principles 20M14 Souslin Tree PFA approachability ideal P-Ideal Dichotomy projective Boolean algebra Aronszajn tree polarized partition relation weak diamond Whitehead Problem Singular cardinals combinatorics Large Cardinals Successor of Singular Cardinal weak square Fast club Constructible Universe Kurepa Hypothesis Erdos Cardinal Cohen real Mandelbrot set Singular coﬁnality Cardinal Invariants Diamond Shelah's Strong Hypothesis Microscopic Approach Jonsson cardinal Antichain Foundations coloring number Martin's Axiom S-Space Club Guessing Hereditarily Lindelöf space Generalized Clubs 05D10 Almost Souslin Slim tree b-scale free Boolean algebra Square-Brackets Partition Relations Non-saturation Reduced Power Hedetniemi's conjecture Successor of Regular Cardinal Small forcing Erdos-Hajnal graphs L-space Uniformization Knaster Parameterized proxy principle Partition Relations Almost-disjoint famiy Singular Cofinality Ascent Path xbox Hindman's Theorem PFA(S)[S] Minimal Walks tensor product graph stationary reflection square Rock n' Roll Rainbow sets very good scale ccc Prevalent singular cardinals incompactness 11P99 Ostaszewski square Poset Fodor-type reflection OCA 05A17 Chang's conjecture diamond star Stevo Todorcevic Coherent tree sap middle diamond stationary hitting Weakly compact cardinal Axiom R Universal Sequences Singular Density

# Tag Archives: sap

## 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