  Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and
model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is
known that in contrast with the classes of all projective and flat modules, the
class of all flat Mittag-Leffler modules is not deconstructible [14], and it
does not provide for approximations when $R$ has cardinality $\leq \aleph_0$,
[6]. We remove the cardinality restriction on $R$ in the latter result. We also
prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair
$(\mathcal A,\mathcal B)$ is of countable type whenever the class $\mathcal B$
is closed under direct limits.
  In order to prove these results, we develop new general tools combining
relative Mittag-Leffler conditions with set-theoretic homological algebra. They
make it possible to trace the facts above to their ultimate, countable, origins
in the properties of Bass modules. These tools have already found a number of
applications: e.g., they yield a positive answer to Enochs' problem on module
approximations for classes of modules associated with tilting [4], and enable
investigation of new classes of flat modules occurring in algebraic geometry
[24]. Finally, the ideas from Section 3 have led to the solution of a
long-standing problem due to Auslander on the existence of right almost split
maps [20].
