Sad news: Jim Baumgartner passed away. See here.
Archives
Keywords
countably metacompact Greatly Mahlo Parameterized proxy principle C-sequence Commutative cancellative semigroups Sierpinski's onto mapping principle transformations Postprocessing function Nonspecial tree Was Ulam right b-scale Forcing Axioms O-space Ramsey theory over partitions Successor of Regular Cardinal Knaster and friends Sigma-Prikry Generalized descriptive set theory Luzin set Almost-disjoint family Distributive tree Ulam matrix P-Ideal Dichotomy specializable Souslin tree tensor product graph Precaliber approachability ideal Coherent tree stationary reflection unbounded function Mandelbrot set Axiom R Antichain super-Souslin tree Small forcing xbox Amenable C-sequence Slim tree Reflecting stationary set Partition Relations L-space Local Club Condensation. Universal Sequences Diamond Open Access Erdos Cardinal Jonsson cardinal projective Boolean algebra positive partition relation Uniformly homogeneous Microscopic Approach Chang's conjecture coloring number Cohen real AIM forcing square principles polarized partition relation Successor of Singular Cardinal Prevalent singular cardinals Shelah's Strong Hypothesis Hereditarily Lindelöf space OCA square regressive Souslin tree Poset Constructible Universe weak Kurepa tree Vanishing levels Uniformly coherent Cardinal Invariants Knaster Well-behaved magma Fat stationary set Martin's Axiom Weakly compact cardinal weak diamond SNR Almost countably chromatic Dowker space Rado's conjecture Large Cardinals Ascent Path Fast club Subadditive weak square Analytic sets club_AD Minimal Walks Singular cofinality Chromatic number Absoluteness Subnormal ideal S-Space Sakurai's Bell inequality Singular cardinals combinatorics nonmeager set Iterated forcing Subtle tree property Almost Souslin Square-Brackets Partition Relations Rock n' Roll reflection principles Prikry-type forcing Subtle cardinal Whitehead Problem Diamond-sharp GMA Aronszajn tree PFA(S)[S] Kurepa Hypothesis Ostaszewski square Strongly Luzin set Forcing Club Guessing Singular Density higher Baire space Fodor-type reflection Erdos-Hajnal graphs Selective Ultrafilter stationary hitting middle diamond Rainbow sets diamond star free Souslin tree PFA Lipschitz reduction full tree Closed coloring Hindman's Theorem Generalized Clubs Non-saturation very good scale strongly bounded groups Souslin Tree Uniformization Diamond for trees free Boolean algebra stick sap Reduced Power Hedetniemi's conjecture incompactness HOD Dushnik-Miller indecomposable ultrafilter 54G20 Ineffable cardinal Cardinal function ccc ZFC construction Strong coloring Filter reflection Foundations
That’s very sad news indeed.
That is sad.
(*) I will always associate Jim with the phrase “Just Baumgartner it!”. Simon Thomas’s word for bringing in an ultrafilter, seemingly for no reason, and letting it guide your decisions.
(**) I would also like to note that Jim shares his first two names with the actor who portrayed Darth Vader.
This is very sad news indeed. The email sent around Dartmouth said that he passed away while his whole family was visiting for the holidays, so Jim was fortunate to spend his last moments surrounded by family and high spirits.
Baumgartner proved that, in contrast to (weak) square, partial square may hold at a singular cardinal (of any cofinality) above a supercomapct carinal. He also proved that after collapsing a weakly comapct cardinal to $omega_2$, every statinoary subset of $S^2_0$ reflects.
These two gems together with an additional argument shows that $square^*_{aleph_{omega_1}}$ may be introduced by a cofinality-preserving small forcing.