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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

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

# 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