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