  In this article, we consider the notion of almost irredundant sets: A subset
$\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and
only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the
norm-closure of $$\{\sum_{i=1}^n \lambda_i \prod_{j=1}^{n_i}a_{{i,j}}: \textrm{
where } a_{{i,j}} \in \mathcal{X}\setminus\{a\} \textrm{ and} \sum
|\lambda_i|\leq 1\}.$$ Since every almost irrredundant set is in particular a
discrete set, it follows that the density of $\mathcal{A}$ is an upper bound
for the size of almost irredundant sets. We prove that under the Proper Forcing
Axiom (PFA), there is an uncountable almost irredundant set in every C*-algebra
with an uncountable increasing sequence of ideals. In particular, assuming PFA,
every nonseparable scattered C*-algebra admits an uncountable almost
irredundant set.
