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