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weak Kurepa tree weak diamond S-Space Shelah's Strong Hypothesis square Fodor-type reflection Diamond for trees sap Reflecting stationary set Partition Relations positive partition relation Diamond stationary reflection incompactness Local Club Condensation. Large Cardinals C-sequence Analytic sets Foundations Subtle cardinal Entangled linear order middle diamond Luzin set Well-behaved magma Commutative projection system Commutative cancellative semigroups Forcing Square-Brackets Partition Relations Distributive tree Absoluteness Respecting tree super-Souslin tree PFA Ascending path specializable Souslin tree Ineffable cardinal Open Access Chromatic number Forcing Axioms Almost Souslin Almost-disjoint family Universal Sequences O-space Erdos Cardinal Mandelbrot set perfectly normal Subadditive tensor product graph stationary hitting Martin's Axiom Closed coloring Constructible Universe Minimal Walks Weakly compact cardinal Cardinal Invariants Cardinal function Small forcing Souslin Tree Sakurai's Bell inequality Microscopic Approach Erdos-Hajnal graphs b-scale Club Guessing Rainbow sets Poset Forcing with side conditions ccc higher Baire space projective Boolean algebra nonmeager set Monotonically far Precaliber Prevalent singular cardinals Iterated forcing Filter reflection Greatly Mahlo 54G20 P-Ideal Dichotomy Generalized Clubs Reduced Power Successor of Regular Cardinal Uniformly homogeneous Uniformization Subnormal ideal GMA Almost countably chromatic Chang's conjecture Ascent Path coloring number regressive Souslin tree xbox weak square Singular Density OCA Singular cofinality Kurepa Hypothesis Singular cardinals combinatorics indecomposable filter Aronszajn tree Strongly compact cardinal club_AD Knaster and friends Parameterized proxy principle Countryman line Vanishing levels Antichain strongly bounded groups Generalized descriptive set theory ZFC construction Strongly Luzin set Sigma-Prikry Ramsey theory over partitions Intersection model free Boolean algebra Rado's conjecture transformations Hindman's Theorem square principles reflection principles Was Ulam right? Fast club SNR Coherent tree Hereditarily Lindelöf space Slim tree free Souslin tree Jonsson cardinal very good scale PFA(S)[S] Amenable C-sequence Prikry-type forcing Subtle tree property L-space AIM forcing Strong coloring diamond star Nonspecial tree unbounded function Selective Ultrafilter HOD Cohen real countably metacompact Interval topology on trees Diamond-sharp Whitehead Problem Dowker space Dushnik-Miller Ulam matrix Knaster Rock n' Roll Successor of Singular Cardinal stick Uniformly coherent Lipschitz reduction Non-saturation Ostaszewski square full tree Sierpinski's onto mapping principle Axiom R polarized partition relation Hedetniemi's conjecture Postprocessing function Fat stationary set approachability ideal Partition relations for trees
Tag Archives: Forcing Axioms
RIMS workshop on Set Theory 2025
I gave an online contributed talk at the RIMS workshop on Set Theory 2025 in Kyoto, December 2025. Talk Title: What is a higher forcing axiom? Abstract: There are multiple interpretations of what is a forcing axiom. We shall survey … Continue reading
The 18th International Workshop on Set Theory in Luminy, November 2015
I gave an invited talk at the 18th International Workshop on Set Theory in Luminy in Marseille, November 2025. Talk Title: What is a higher forcing axiom? Abstract: There are multiple interpretations of what is a forcing axiom. We shall … Continue reading
Posted in Invited Talks
Tagged Forcing Axioms
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Squares, ultrafilters and forcing axioms
Joint work with Chris Lambie-Hanson and Jing Zhang. Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following. Strong forcing axioms, in general incompatible with the existence of indexed squares, can … 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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Generalizations of Martin’s Axiom and the well-met condition
Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading
Posted in Blog, Expository
Tagged ccc, Forcing Axioms, GMA, Martin's Axiom, Uniformization
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Bell’s theorem on the cardinal invariant $\mathfrak p$
In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, … Continue reading
