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middle diamond Microscopic Approach Fodor-type reflection Axiom R OCA Ascent Path Strongly compact cardinal Prikry-type forcing Commutative projection system Rado's conjecture PFA Square-Brackets Partition Relations Erdos Cardinal Interval topology on trees Uniformly homogeneous Generalized descriptive set theory Subadditive incompactness very good scale Postprocessing function Generalized Clubs Reduced Power Shelah's Strong Hypothesis stationary reflection approachability ideal Greatly Mahlo Hindman's Theorem Diamond Almost countably chromatic Club Guessing Chromatic number Strong coloring Local Club Condensation. S-Space Foundations Hereditarily Lindelöf space transformations perfectly normal weak Kurepa tree free Boolean algebra Singular cardinals combinatorics club_AD Well-behaved magma unbounded function Intersection model polarized partition relation square sap countably metacompact L-space Slim tree Filter reflection Weakly compact cardinal Absoluteness Poset Coherent tree full tree Fast club Singular Density Dowker space Entangled linear order Forcing PFA(S)[S] Kurepa Hypothesis Almost Souslin Almost-disjoint family Selective Ultrafilter Successor of Singular Cardinal Commutative cancellative semigroups specializable Souslin tree Diamond for trees Open Access ccc Universal Sequences regressive Souslin tree Cohen real ZFC construction b-scale Sigma-Prikry Forcing Axioms diamond star stick Was Ulam right? square principles C-sequence Vanishing levels strongly bounded groups Uniformly coherent Non-saturation Strongly Luzin set P-Ideal Dichotomy Cardinal Invariants stationary hitting Partition relations for trees Jonsson cardinal Rainbow sets Chang's conjecture Analytic sets Mandelbrot set Monotonically far weak diamond tensor product graph reflection principles super-Souslin tree Ascending path 54G20 xbox Martin's Axiom Sierpinski's onto mapping principle Successor of Regular Cardinal projective Boolean algebra Distributive tree Ineffable cardinal coloring number Reflecting stationary set SNR Precaliber Respecting tree Aronszajn tree O-space free Souslin tree Knaster Souslin Tree Erdos-Hajnal graphs Prevalent singular cardinals Partition Relations Dushnik-Miller Cardinal function Iterated forcing Constructible Universe Ramsey theory over partitions Luzin set Diamond-sharp Subtle tree property Nonspecial tree HOD Singular cofinality Amenable C-sequence Whitehead Problem Forcing with side conditions Subnormal ideal positive partition relation Countryman line Fat stationary set Sakurai's Bell inequality Antichain Parameterized proxy principle Closed coloring Large Cardinals Knaster and friends Ostaszewski square Rock n' Roll nonmeager set Hedetniemi's conjecture Minimal Walks Small forcing GMA AIM forcing indecomposable filter Subtle cardinal weak square higher Baire space Uniformization Ulam matrix Lipschitz reduction
Tag Archives: 03E05
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Answering a question of Ketonen from the late 1970’s, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing … Continue reading
Fake Reflection
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
A remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
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A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
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Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
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