### 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

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

# 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