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### 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

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

# Tag Archives: ccc

## 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

## Chain conditions of products, and weakly compact cardinals

Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading

Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
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## c.c.c. vs. the Knaster property

After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … 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