  The tower number $\mathfrak t$ and the ultrafilter number $\mathfrak u$ are
cardinal characteristics from set theory. They are based on combinatorial
properties of classes of subsets of~$\omega$ and the almost inclusion relation
$\subseteq^*$ between such subsets. We consider analogs of these cardinal
characteristics in computability theory.
  We show that the mass problem of ultrafilter bases is equivalent to the mass
problem of computing a function that dominates all computable functions, and
hence, by Martin's characterization, it captures highness. On the other hand,
the mass problem for maximal towers is below the mass problem of computing a
non-low set. We also show that some, but not all, noncomputable low sets
compute maximal towers: Every noncomputable (low) c.e.\ set computes a maximal
tower but no 1-generic $\Delta^0_2$-set does so.
  We finally consider the mass problems of maximal almost disjoint, and of
maximal independent families. We show that they are Medvedev equivalent to
maximal towers, and to ultrafilter bases, respectively.
