# Tag Archives: Uniformization

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 $G\cap … Continue reading Posted in Blog, Expository | Tagged , , | Leave a comment ## The uniformization property for$\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)$is a … Continue reading Posted in Blog, Expository | Tagged | Leave a comment ## c.c.c. forcing without combinatorics In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing$\mathbb P\$ is said to satisfy the c.c.c. iff … Continue reading

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## 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