Archives
Keywords
Microscopic Approach stick weak diamond Prevalent singular cardinals b-scale C-sequence Ineffable cardinal weak Kurepa tree Prikry-type forcing Absoluteness Diamond approachability ideal Selective Ultrafilter transformations Knaster Subtle tree property Hereditarily Lindelöf space Greatly Mahlo positive partition relation Foundations square ZFC construction perfectly normal HOD Rainbow sets coloring number Cohen real Uniformly homogeneous Aronszajn tree Slim tree free Boolean algebra Generalized Clubs very good scale Filter reflection Commutative projection system Non-saturation square principles Souslin Tree 54G20 diamond star Sigma-Prikry Monotonically far tensor product graph stationary reflection Postprocessing function Subtle cardinal reflection principles specializable Souslin tree Sakurai's Bell inequality Mandelbrot set Erdos-Hajnal graphs Cardinal function Square-Brackets Partition Relations Chromatic number Analytic sets Partition Relations Rock n' Roll Weakly compact cardinal countably metacompact Universal Sequences Diamond-sharp indecomposable filter PFA Ostaszewski square Subnormal ideal Sierpinski's onto mapping principle Dowker space Was Ulam right? unbounded function Partition relations for trees Strongly Luzin set Fast club Coherent tree PFA(S)[S] super-Souslin tree S-Space P-Ideal Dichotomy Singular cardinals combinatorics Uniformization Small forcing OCA Chang's conjecture stationary hitting Precaliber Erdos Cardinal Forcing Kurepa Hypothesis ccc Reflecting stationary set Ulam matrix Respecting tree Generalized descriptive set theory Large Cardinals Successor of Regular Cardinal Ascent Path Singular cofinality Successor of Singular Cardinal O-space full tree Almost countably chromatic Distributive tree nonmeager set Entangled linear order Commutative cancellative semigroups Ramsey theory over partitions projective Boolean algebra AIM forcing Uniformly coherent Open Access Vanishing levels Club Guessing Antichain Parameterized proxy principle regressive Souslin tree Dushnik-Miller higher Baire space Hedetniemi's conjecture Almost Souslin Luzin set Well-behaved magma incompactness Whitehead Problem club_AD Singular Density xbox Hindman's Theorem Jonsson cardinal GMA Nonspecial tree Axiom R Ascending path polarized partition relation weak square Countryman line Subadditive Diamond for trees Forcing with side conditions Rado's conjecture Lipschitz reduction L-space free Souslin tree Fat stationary set Amenable C-sequence Constructible Universe Iterated forcing Interval topology on trees Shelah's Strong Hypothesis Cardinal Invariants Reduced Power Almost-disjoint family Martin's Axiom strongly bounded groups Strongly compact cardinal Local Club Condensation. middle diamond Intersection model Fodor-type reflection Closed coloring Minimal Walks Poset SNR sap Strong coloring Forcing Axioms Knaster and friends
Tag Archives: xbox
Proxy principles in combinatorial set theory
Joint work with Ari Meir Brodsky and Shira Yadai. Abstract. The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not … Continue reading
Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
2 Comments
A microscopic approach to Souslin-tree constructions. Part II
Joint work with Ari Meir Brodsky. Abstract. In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of … Continue reading
More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
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
16 Comments
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
5 Comments
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
1 Comment
Square principles
Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading
