### 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

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

# 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