### 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

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

# 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