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