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