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