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