### 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

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

# Tag Archives: Uniformization

## Generalizations of Martin’s Axiom and the well-met condition

Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading

## The uniformization property for $\aleph_2$

Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a … Continue reading

## c.c.c. forcing without combinatorics

In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing $\mathbb P$ is said to satisfy the c.c.c. iff … Continue reading

## 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

## On guessing generalized clubs at the successors of regulars

Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading