### 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

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

# 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