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