# Tag Archives: ccc

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 $G\cap … Continue reading Posted in Blog, Expository | Tagged , , | Leave a comment ## Chain conditions of products, and weakly compact cardinals Abstract. The history of productivity of the$\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading Posted in Partition Relations, Publications | | 2 Comments ## c.c.c. vs. the Knaster property After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … Continue reading Posted in Blog | Tagged , , , | Leave a comment ## c.c.c. forcing without combinatorics In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing$\mathbb P\$ is said to satisfy the c.c.c. iff … Continue reading

Posted in Blog, Expository | Tagged , | 5 Comments