  In this paper we are concerned with the problem of completeness in the
Bergman space of the worm domain $\mathcal{W}_\mu$ and its truncated version
$\mathcal{W}'_\mu$. We determine some orthogonal systems and show that they are
not complete, while showing that the union of two particular of such systems is
complete.
  In order to prove our completeness result we introduce the Muentz-Szasz
problem for the 1-dimensional Bergman space of the disk $\{\zeta :
|\zeta-1|<1\}$ and find a sufficient condition for its solution.
