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

Maximal towers and ultrafilter bases in computability