### 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

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

# 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