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Subnormal ideal b-scale Dowker space Amenable C-sequence Axiom R Constructible Universe Successor of Regular Cardinal L-space Partition Relations Knaster and friends Cohen real indecomposable filter Respecting tree Selective Ultrafilter Cardinal function S-Space Rado's conjecture Sigma-Prikry super-Souslin tree stationary hitting Absoluteness HOD Fodor-type reflection Chang's conjecture Uniformly homogeneous unbounded function very good scale diamond star Sakurai's Bell inequality incompactness countably metacompact Shelah's Strong Hypothesis Diamond for trees Souslin Tree Strong coloring Singular cofinality projective Boolean algebra Rainbow sets O-space Foundations Reduced Power square principles Well-behaved magma Ascending path Chromatic number Minimal Walks Erdos Cardinal Diamond strongly bounded groups Singular cardinals combinatorics tensor product graph Cardinal Invariants Strongly Luzin set Hereditarily Lindelöf space Coherent tree Intersection model Mandelbrot set square Universal Sequences Prevalent singular cardinals nonmeager set Interval topology on trees weak Kurepa tree Hedetniemi's conjecture Knaster Poset Forcing with side conditions PFA Slim tree Antichain P-Ideal Dichotomy Whitehead Problem Ramsey theory over partitions weak square Small forcing stick reflection principles Subtle cardinal AIM forcing Successor of Singular Cardinal Fat stationary set sap Forcing Vanishing levels Club Guessing Singular Density approachability ideal perfectly normal Almost countably chromatic full tree middle diamond Parameterized proxy principle Local Club Condensation. specializable Souslin tree Filter reflection Almost-disjoint family Lipschitz reduction Hindman's Theorem Postprocessing function Ineffable cardinal Non-saturation Sierpinski's onto mapping principle OCA weak diamond Nonspecial tree Greatly Mahlo Was Ulam right? Generalized descriptive set theory Jonsson cardinal 54G20 Large Cardinals Aronszajn tree PFA(S)[S] Prikry-type forcing ZFC construction polarized partition relation Forcing Axioms Generalized Clubs Precaliber Ostaszewski square GMA Dushnik-Miller Subtle tree property Kurepa Hypothesis Fast club xbox Closed coloring transformations Uniformization Weakly compact cardinal Almost Souslin Partition relations for trees Ascent Path coloring number Rock n' Roll Monotonically far free Souslin tree positive partition relation Erdos-Hajnal graphs Ulam matrix Square-Brackets Partition Relations Commutative projection system Reflecting stationary set Countryman line Subadditive Entangled linear order Open Access Commutative cancellative semigroups Uniformly coherent regressive Souslin tree stationary reflection Distributive tree Iterated forcing Analytic sets SNR higher Baire space Diamond-sharp Microscopic Approach club_AD Martin's Axiom Luzin set ccc C-sequence free Boolean algebra Strongly compact cardinal
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
