Archives
Keywords
Greatly Mahlo Large Cardinals Respecting tree Subadditive Ascent Path Universal Sequences stationary reflection middle diamond Coherent tree Subnormal ideal perfectly normal Almost-disjoint family Club Guessing Singular Density square Was Ulam right? Generalized descriptive set theory 54G20 free Boolean algebra S-Space positive partition relation stationary hitting Microscopic Approach club_AD AIM forcing Ulam matrix Strong coloring Chromatic number Subtle cardinal Ramsey theory over partitions Almost countably chromatic Strongly Luzin set regressive Souslin tree b-scale Postprocessing function Subtle tree property polarized partition relation Singular cardinals combinatorics Commutative projection system Intersection model Antichain Erdos-Hajnal graphs indecomposable filter Knaster and friends Sigma-Prikry Rainbow sets sap Prevalent singular cardinals SNR higher Baire space Fodor-type reflection strongly bounded groups Diamond for trees Uniformly coherent Jonsson cardinal Iterated forcing Interval topology on trees Kurepa Hypothesis Small forcing Reduced Power Weakly compact cardinal O-space Rock n' Roll Absoluteness Cohen real OCA weak diamond Constructible Universe Nonspecial tree C-sequence Strongly compact cardinal full tree Sakurai's Bell inequality Countryman line Analytic sets Cardinal Invariants Lipschitz reduction Forcing Axioms L-space unbounded function Precaliber Chang's conjecture Cardinal function Diamond Parameterized proxy principle stick Amenable C-sequence Hereditarily Lindelöf space Foundations approachability ideal Forcing coloring number very good scale Slim tree Forcing with side conditions Uniformization Axiom R free Souslin tree Selective Ultrafilter transformations Dowker space Martin's Axiom Vanishing levels Prikry-type forcing Rado's conjecture countably metacompact GMA nonmeager set projective Boolean algebra Shelah's Strong Hypothesis Partition Relations Closed coloring HOD Reflecting stationary set ccc Poset Ascending path Dushnik-Miller reflection principles Aronszajn tree incompactness Ineffable cardinal Minimal Walks square principles Whitehead Problem Knaster diamond star Ostaszewski square weak Kurepa tree Almost Souslin Monotonically far super-Souslin tree Distributive tree Generalized Clubs P-Ideal Dichotomy Entangled linear order Successor of Regular Cardinal Partition relations for trees Uniformly homogeneous Fat stationary set PFA(S)[S] Open Access Erdos Cardinal Well-behaved magma Hedetniemi's conjecture Successor of Singular Cardinal specializable Souslin tree weak square Commutative cancellative semigroups xbox Filter reflection PFA Non-saturation Luzin set tensor product graph Square-Brackets Partition Relations Souslin Tree Mandelbrot set Hindman's Theorem Sierpinski's onto mapping principle Local Club Condensation. ZFC construction Fast club Singular cofinality Diamond-sharp
Tag Archives: Cardinal Invariants
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
1 Comment
Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Cardinal Invariants, Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
4 Comments
Jones’ theorem on the cardinal invariant $\mathfrak p$
This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading
Bell’s theorem on the cardinal invariant $\mathfrak p$
In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, … Continue reading
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Infinite Combinatorial Topology
Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading
Posted in Notes
Tagged b-scale, Cardinal function, Cardinal Invariants, Hereditarily Lindelöf space
8 Comments
