  A subset $\mathcal X$ of a C*-algebra $\mathcal A$ is called irredundant if
no $A\in \mathcal X$ belongs to the C*-subalgebra of $\mathcal A$ generated by
$\mathcal X\setminus \{A\}$. Separable C*-algebras cannot have uncountable
irredundant sets and all members of many classes of nonseparable C*-algebras,
e.g., infinite dimensional von Neumann algebras have irredundant sets of
cardinality continuum.
  There exists a considerable literature showing that the question whether
every AF commutative nonseparable C*-algebra has an uncountable irredundant set
is sensitive to additional set-theoretic axioms and we investigate here the
noncommutative case.
  Assuming $\diamondsuit$ (an additional axiom stronger than the continuum
hypothesis) we prove that there is an AF C*-subalgebra of $\mathcal B(\ell_2)$
of density $2^\omega=\omega_1$ with no nonseparable commutative C*-subalgebra
and with no uncountable irredundant set. On the other hand we also prove that
it is consistent that every discrete collection of operators in $\mathcal
B(\ell_2)$ of cardinality continuum contains an irredundant subcollection of
cardinality continuum.
  Other partial results and more open problems are presented.

Operator Algebras;General Topology;Logic