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OCA Cohen real weak Kurepa tree Reflecting stationary set Forcing with side conditions stationary hitting regressive Souslin tree Respecting tree Singular cofinality S-Space Fast club Subnormal ideal countably metacompact Ascending path sap Forcing Axioms Almost countably chromatic weak diamond incompactness Entangled linear order Sakurai's Bell inequality Fodor-type reflection Square-Brackets Partition Relations Axiom R Rainbow sets transformations Absoluteness Poset specializable Souslin tree Was Ulam right? SNR Reduced Power higher Baire space Parameterized proxy principle square principles Jonsson cardinal Ulam matrix Ostaszewski square Constructible Universe Shelah's Strong Hypothesis full tree Weakly compact cardinal super-Souslin tree ccc Cardinal function Antichain Fat stationary set polarized partition relation Forcing Microscopic Approach Knaster and friends stick club_AD Strong coloring Strongly compact cardinal Rado's conjecture Hedetniemi's conjecture Non-saturation square Slim tree Sigma-Prikry Filter reflection Prevalent singular cardinals Luzin set Almost Souslin 54G20 Coherent tree Almost-disjoint family Generalized Clubs Vanishing levels Commutative cancellative semigroups Foundations indecomposable filter reflection principles Ascent Path Prikry-type forcing Well-behaved magma Subtle tree property Aronszajn tree Sierpinski's onto mapping principle AIM forcing Analytic sets Diamond for trees Uniformly coherent Singular Density Iterated forcing Whitehead Problem Selective Ultrafilter Partition relations for trees Countryman line Small forcing Distributive tree PFA Diamond Erdos-Hajnal graphs Intersection model approachability ideal Universal Sequences free Boolean algebra ZFC construction b-scale Generalized descriptive set theory O-space weak square Cardinal Invariants Dowker space Subtle cardinal Monotonically far Uniformly homogeneous P-Ideal Dichotomy diamond star Club Guessing Nonspecial tree Interval topology on trees Postprocessing function Successor of Singular Cardinal Singular cardinals combinatorics middle diamond Chang's conjecture Chromatic number Souslin Tree coloring number Large Cardinals Successor of Regular Cardinal PFA(S)[S] Closed coloring tensor product graph strongly bounded groups Ramsey theory over partitions Subadditive Open Access Hereditarily Lindelöf space Erdos Cardinal Knaster Precaliber Martin's Axiom stationary reflection GMA Diamond-sharp Commutative projection system Lipschitz reduction Partition Relations Rock n' Roll Kurepa Hypothesis positive partition relation Minimal Walks Uniformization Amenable C-sequence C-sequence Local Club Condensation. free Souslin tree unbounded function L-space Dushnik-Miller xbox Mandelbrot set Hindman's Theorem very good scale Strongly Luzin set HOD nonmeager set Ineffable cardinal Greatly Mahlo projective Boolean algebra perfectly normal
Category Archives: Blog
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
Afghan Whigs on Jimmy Fallon
Performing “I’m Her Slave” (from their album Congregation) at NBC’s studios, 22-May-2012:
The chromatic numbers of the Erdos-Hajnal graphs
Recall that a coloring $c:G\rightarrow\kappa$ of an (undirected) graph $(G,E)$ is said to be chromatic if $c(v_1)\neq c(v_2)$ whenever $\{v_1,v_2\}\in E$. Then, the chromatic number of a graph $(G,E)$ is the least cardinal $\kappa$ for which there exists a chromatic … Continue reading
Posted in Blog, Expository
Tagged Chromatic number, Erdos-Hajnal graphs, Rado's conjecture, reflection principles
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Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … Continue reading
Review: Is classical set theory compatible with quantum experiments?
Yesterday, I attended a talk at the Quantum Foundations seminar at the beautiful Perimeter Institute for Theoretical Physics (Waterloo, Ontario). The (somewhat provocative) title of the talk was “Is Classical Set Theory Compatible with Quantum Experiments?”, and the speaker was Radu … Continue reading
Comparing rectangles with squares through rainbow sets
In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading
