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Dispute over Infinity Divides Mathematicians
Scientific American online
But unlike most of the ZFC axioms, the new ones “are not self-evident, or at least not self-evident at this stage of our knowledge, so we have a much more difficult task,” said Stevo Todorcevic, a mathematician at the University of Toronto and the French National Center for Scientific Research in Paris.
An n-gap consists of n many pairwise orthogonal families of subsets of a countable set that cannot be separated. We prove that for every positive integer n there is a finite basis for the class of analytic n-gaps. The proof requires an analysis of certain combinatorial problems on the n-adic tree, and in particular a new partition theorem for trees...
Motivated by a Tukey classification problem, we develop a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space. Associated with Ra is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1...