  Given a finite permutation group $G$ with domain $\Omega$, we associate two
subsets of natural numbers to $G$, namely $\mathcal{I}(G,\Omega)$ and
$\mathcal{M}(G,\Omega)$, which are the sets of cardinalities of all the
irredundant and minimal bases of $G$, respectively. We prove that
$\mathcal{I}(G)$ is an interval of natural numbers, whereas
$\mathcal{M}(G,\Omega)$ may not necessarily form an interval. Moreover, for a
given subset of natural numbers $X \subseteq \mathbb{N}$, we provide some
conditions on $X$ that ensure the existence of both intransitive and transitive
groups $G$ such that
  $\mathcal{I}(G,\Omega) = X$ and $\mathcal{M}(G,\Omega) = X$.
