CMS winter meeting, Toronto, December 2011. (Toronto slides)
Abstract: Many proofs of inequalities between cardinal characteristics of the continuum are combinatorial in nature. These arguments can be carried out in any model of set theory, even a model of CH where the inequalities themselves are trivial. Thus, such arguments appear to establish a stronger relationship than a mere inequality. The Borel Tukey order was introduced by Blass in a 1996 article to address just this. Specifically, he observed that the combinatorial information linking two cardinal characteristics is often captured by a pair of Borel maps called a Borel Tukey morphism. The existence of a Borel Tukey morphisms between two cardinal invariants has since been found to have a couple of applications in other set-theoretic contexts. In this talk we will discuss a number of popular combinatorial cardinal invariants, and compare their traditional ordering of provable inequalities with the finer ordering given by the Borel Tukey morphisms.