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