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