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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
- Genearlizations 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

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

# Tag Archives: ccc

## Genearlizations 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