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Constructible Universe Whitehead Problem Analytic sets Diamond Countryman line Distributive tree Successor of Singular Cardinal Ulam matrix Almost countably chromatic Diamond for trees Commutative cancellative semigroups weak diamond Rock n' Roll ZFC construction Cohen real unbounded function P-Ideal Dichotomy incompactness Intersection model Commutative projection system Chang's conjecture coloring number Microscopic Approach Well-behaved magma Dowker space Club Guessing regressive Souslin tree Singular Density Forcing Strong coloring countably metacompact nonmeager set Fast club Subadditive Dushnik-Miller Ascent Path Precaliber Universal Sequences Vanishing levels Ramsey theory over partitions club_AD Partition Relations Sierpinski's onto mapping principle Chromatic number Hindman's Theorem Singular cardinals combinatorics ccc Reflecting stationary set Absoluteness Parameterized proxy principle Strongly compact cardinal Filter reflection Subnormal ideal stationary hitting free Souslin tree HOD Generalized descriptive set theory Non-saturation Square-Brackets Partition Relations weak square Minimal Walks Cardinal Invariants Iterated forcing Coherent tree 54G20 PFA(S)[S] Sigma-Prikry Jonsson cardinal Uniformly homogeneous Successor of Regular Cardinal Weakly compact cardinal L-space Ineffable cardinal middle diamond stick Hedetniemi's conjecture Generalized Clubs Selective Ultrafilter free Boolean algebra strongly bounded groups Large Cardinals Open Access Amenable C-sequence Closed coloring Reduced Power Rainbow sets Foundations stationary reflection OCA Diamond-sharp Sakurai's Bell inequality Aronszajn tree Greatly Mahlo Slim tree Singular cofinality Forcing Axioms approachability ideal S-Space weak Kurepa tree SNR Shelah's Strong Hypothesis full tree higher Baire space Postprocessing function positive partition relation Cardinal function Poset indecomposable ultrafilter polarized partition relation Mandelbrot set xbox Uniformly coherent Souslin Tree GMA Luzin set Fat stationary set sap specializable Souslin tree Kurepa Hypothesis Local Club Condensation. Knaster diamond star Nonspecial tree Rado's conjecture O-space Strongly Luzin set reflection principles projective Boolean algebra Respecting tree Erdos-Hajnal graphs Lipschitz reduction Subtle tree property tensor product graph Prikry-type forcing Knaster and friends Antichain square principles transformations Uniformization b-scale Hereditarily Lindelöf space Erdos Cardinal AIM forcing Almost Souslin Ostaszewski square Subtle cardinal super-Souslin tree C-sequence Martin's Axiom square Fodor-type reflection very good scale PFA Was Ulam right Small forcing Axiom R Almost-disjoint family Prevalent singular cardinals
Category Archives: Blog
What’s next?
I took an offer for a tenure-track position at the Mathematics department of Bar-Ilan University.
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Review: Stevo Todorcevic’s CRM-Fields-PIMS Prize Lecture
After winning the 2012 CRM-Fields-PIMS Prize, Stevo Todorcevic gave a series of talks on his research: at CRM, at PIMS and at the Fields Institute. The director of the Fields Institute asked me to write a short review on Stevo’s … Continue reading
Prikry Forcing
Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is … Continue reading
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
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Tagged approachability ideal, Club Guessing
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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
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 Engelking-Karlowicz theorem, and a useful corollary
Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in[\mu]^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading
Kurepa trees and ineffable cardinals
Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. … Continue reading
Variations on diamond
Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little … Continue reading
The P-Ideal Dichotomy and the Souslin Hypothesis
John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading
