### Archives

### Recent blog posts

- 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
- Walk on countable ordinals: the characteristics December 1, 2013

### Keywords

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

# 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