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strongly bounded groups S-Space specializable Souslin tree Closed coloring coloring number Singular cardinals combinatorics Coherent tree unbounded function Hereditarily Lindelöf space higher Baire space GMA b-scale Successor of Singular Cardinal Greatly Mahlo Almost Souslin Strong coloring Whitehead Problem free Souslin tree regressive Souslin tree weak square Square-Brackets Partition Relations Subadditive P-Ideal Dichotomy Diamond for trees tensor product graph Chromatic number Postprocessing function Diamond weak diamond Prevalent singular cardinals Diamond-sharp full tree Was Ulam right Sakurai's Bell inequality diamond star Slim tree Uniformly homogeneous Club Guessing Non-saturation Knaster and friends Local Club Condensation. Generalized descriptive set theory PFA Uniformization Open Access nonmeager set Luzin set Martin's Axiom OCA Aronszajn tree Weakly compact cardinal Knaster HOD Antichain Successor of Regular Cardinal Axiom R Souslin Tree Reduced Power Ulam matrix Dowker space middle diamond Subtle tree property Cohen real Subnormal ideal Shelah's Strong Hypothesis Almost countably chromatic PFA(S)[S] Rado's conjecture Partition Relations Fodor-type reflection L-space positive partition relation 54G20 Foundations Dushnik-Miller C-sequence Filter reflection Ineffable cardinal indecomposable ultrafilter ccc square stick Prikry-type forcing Distributive tree SNR Amenable C-sequence Ramsey theory over partitions Forcing O-space sap Fast club polarized partition relation Analytic sets Subtle cardinal stationary reflection Commutative cancellative semigroups Parameterized proxy principle Iterated forcing Erdos-Hajnal graphs Hedetniemi's conjecture Jonsson cardinal Well-behaved magma Microscopic Approach Kurepa Hypothesis Rock n' Roll Nonspecial tree free Boolean algebra Precaliber Strongly Luzin set weak Kurepa tree Forcing Axioms Constructible Universe AIM forcing square principles approachability ideal Lipschitz reduction Sigma-Prikry Rainbow sets Vanishing levels Uniformly coherent club_AD reflection principles Singular Density Erdos Cardinal Singular cofinality Fat stationary set very good scale Small forcing incompactness Poset Absoluteness Sierpinski's onto mapping principle Large Cardinals Universal Sequences transformations Selective Ultrafilter Hindman's Theorem ZFC construction Generalized Clubs Ascent Path Cardinal function Mandelbrot set Cardinal Invariants super-Souslin tree projective Boolean algebra stationary hitting Chang's conjecture xbox countably metacompact Reflecting stationary set Almost-disjoint family Ostaszewski square Minimal Walks
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
