I plan to describe two species of ultrafilters on the set of natural numbers and to speculate about a connection between them. For both species, the central question is whether ZFC can prove the existence of such ultrafilters.

One species is defined in terms of the Tukey ordering of directed sets, but it also admits a more combinatorial definition. The other species is defined in terms of preservation by forcing, but it also admits a combinatorial definition. The two combinatorial definitions, though different, have very similar "flavor", and that leads to my speculations.

I'll present the original definitions as well as the combinatorial equivalents, and then I'll discuss attempts to combine the key properties of the two species. Speaker(s): Andreas Blass (UM)