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