We study the problem of existence and generic existence of
ultrafilters on ω. We prove a conjecture of Jörg Brendle’s
showing that there is an ultrafilter that is countably closed but is
not an ordinal ultrafilter under CH. We also show that Canjar’s
previous partial characterization of the generic existence of
Q-points is the best that can be done. More simply put, there is no
normal cardinal invariant equality that fully characterizes the
generic existence of Q-points. We then sharpen results on generic
existence with the introduction of σ-compact ultrafilters. We
show that the generic existence of said ultrafilters is equivalent
to 𝔡=𝔠. This result, taken along with our result that there
exists a Kσ, non-countably closed ultrafilter under CH,
expands the size of the class of ultrafilters that were known to fit
this description before. From the core of the proof, we get a new
result on the cardinal invariants of the continuum, i.e., the
cofinality of the sets with σ-compact closure is 𝔡.