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