### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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