### 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
- Generalizations 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

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

# 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