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Large Cardinals Universal Sequences Non-saturation sap Whitehead Problem Generalized Clubs Hindman's Theorem countably metacompact Dowker space Postprocessing function Constructible Universe Hereditarily Lindelöf space Ostaszewski square nonmeager set Forcing Axioms Reduced Power Greatly Mahlo GMA Shelah's Strong Hypothesis Amenable C-sequence Antichain Cardinal Invariants weak Kurepa tree Erdos-Hajnal graphs Sigma-Prikry Knaster PFA Diamond Luzin set Slim tree Prikry-type forcing positive partition relation strongly bounded groups Axiom R indecomposable ultrafilter regressive Souslin tree Erdos Cardinal ccc OCA Ulam matrix P-Ideal Dichotomy Singular Density PFA(S)[S] Foundations square principles Hedetniemi's conjecture O-space very good scale Coherent tree Well-behaved magma Successor of Regular Cardinal Fat stationary set Small forcing Ascent Path Ramsey theory over partitions Open Access 54G20 Fodor-type reflection Strong coloring Souslin Tree Sakurai's Bell inequality Subtle tree property higher Baire space Poset Subnormal ideal SNR Chromatic number Forcing Martin's Axiom Jonsson cardinal coloring number Almost Souslin Precaliber AIM forcing Subtle cardinal Closed coloring Was Ulam right HOD polarized partition relation Fast club free Souslin tree Filter reflection Rado's conjecture Weakly compact cardinal Diamond for trees Subadditive Microscopic Approach Sierpinski's onto mapping principle Dushnik-Miller Distributive tree tensor product graph xbox super-Souslin tree Successor of Singular Cardinal Commutative cancellative semigroups Generalized descriptive set theory Absoluteness Local Club Condensation. free Boolean algebra Mandelbrot set Aronszajn tree Diamond-sharp Cardinal function C-sequence S-Space full tree Knaster and friends Singular cardinals combinatorics Cohen real projective Boolean algebra L-space Ineffable cardinal Parameterized proxy principle diamond star Kurepa Hypothesis Uniformization Almost-disjoint family weak diamond specializable Souslin tree Analytic sets Prevalent singular cardinals stick Partition Relations Selective Ultrafilter middle diamond Uniformly homogeneous Rock n' Roll stationary hitting Strongly Luzin set Vanishing levels Singular cofinality Nonspecial tree b-scale Square-Brackets Partition Relations Iterated forcing Minimal Walks square club_AD Uniformly coherent Club Guessing incompactness weak square Rainbow sets stationary reflection approachability ideal Chang's conjecture ZFC construction reflection principles Almost countably chromatic Reflecting stationary set transformations Lipschitz reduction unbounded function
Tag Archives: 03E05
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, square, xbox
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Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading