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