  We discuss the existence of (injectively) universal C*-algebras and prove
that all C*-algebras of density character $\aleph_1$ embed into the Calkin
algebra, $Q(H)$. Together with other results, this shows that each of the
following assertions is relatively consistent with ZFC: (i) $Q(H)$ is a
$2^{\aleph_0}$-universal C*-algebra. (ii) There exists a
$2^{\aleph_0}$-universal C*-algebra, but $Q(H)$ is not
$2^{\aleph_0}$-universal. (iii) A $2^{\aleph_0}$-universal C*-algebra does not
exist. We also prove that it is relatively consistent with ZFC that (iv) there
is no $\aleph_1$-universal nuclear C*-algebra, and that (v) there is no
$\aleph_1$-universal simple nuclear C*-algebra.
