The Calkin algebra is ℵ1-universal

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(0)(aleph)-universal C*-algebra. (ii) There exists a 2(0)(aleph)-universal C*-algebra, but Q(H) is not 2(0)(aleph)-universal, (iii) A 2(0)(aleph)-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.