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