Archives
Keywords
Almost Souslin PFA Subtle tree property Ostaszewski square 54G20 stick very good scale stationary reflection Knaster and friends regressive Souslin tree Club Guessing Prikry-type forcing Rock n' Roll Fat stationary set Square-Brackets Partition Relations diamond star positive partition relation projective Boolean algebra higher Baire space C-sequence Postprocessing function Distributive tree Cardinal Invariants Analytic sets Sigma-Prikry Chang's conjecture Subnormal ideal Universal Sequences Closed coloring Ascent Path full tree stationary hitting Prevalent singular cardinals square principles Microscopic Approach Knaster Partition Relations Successor of Regular Cardinal nonmeager set Cohen real Ramsey theory over partitions specializable Souslin tree Ineffable cardinal Poset Small forcing Selective Ultrafilter weak diamond GMA Uniformly homogeneous Iterated forcing Dowker space super-Souslin tree O-space Singular cardinals combinatorics Successor of Singular Cardinal Subtle cardinal Martin's Axiom AIM forcing Minimal Walks Filter reflection Vanishing levels indecomposable ultrafilter Slim tree L-space Luzin set Axiom R coloring number Local Club Condensation. Antichain P-Ideal Dichotomy Souslin Tree Forcing Axioms Strong coloring unbounded function Erdos Cardinal Lipschitz reduction OCA Rado's conjecture Cardinal function club_AD approachability ideal Non-saturation Foundations countably metacompact Fodor-type reflection strongly bounded groups tensor product graph Uniformization Coherent tree Forcing Hindman's Theorem Singular cofinality incompactness Almost-disjoint family square Reflecting stationary set free Boolean algebra Dushnik-Miller reflection principles Almost countably chromatic Generalized descriptive set theory transformations Aronszajn tree b-scale Subadditive Uniformly coherent Chromatic number Generalized Clubs Hereditarily Lindelöf space polarized partition relation ccc Diamond Parameterized proxy principle Precaliber Reduced Power Open Access xbox sap HOD SNR Large Cardinals middle diamond weak square Mandelbrot set Commutative cancellative semigroups Ulam matrix Erdos-Hajnal graphs Diamond for trees Sierpinski's onto mapping principle Fast club Well-behaved magma Hedetniemi's conjecture Shelah's Strong Hypothesis Amenable C-sequence Kurepa Hypothesis Jonsson cardinal Diamond-sharp Whitehead Problem PFA(S)[S] Nonspecial tree ZFC construction S-Space Strongly Luzin set Rainbow sets Weakly compact cardinal free Souslin tree Absoluteness Greatly Mahlo Constructible Universe Sakurai's Bell inequality Was Ulam right Singular Density
Tag Archives: Aronszajn tree
11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
Leave a comment
The 14th International Workshop on Set Theory in Luminy, October 2017
I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017. Talk Title: Distributive Aronszajn trees Abstract: It is well-known that that the statement “all $\aleph_1$-Aronszajn trees are special” is consistent with ZFC … Continue reading
Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
PFA and the tree property at $\aleph_2$
Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, … Continue reading
A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
Leave a comment