### 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

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

# 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