We demonstrate how virtually all common cardinal invariants associated to a
von Neumann algebra M can be computed from the decomposability number, dec(M),
and the minimal cardinality of a generating set, gen(M). Applications include
the equivalence of the well-known generator problem, "Is every separably-acting
von Neumann algebra singly-generated?", with the formally stronger questions,
"Is every countably-generated von Neumann algebra singly-generated?" and "Is
the gen invariant monotone?" Modulo the generator problem, we determine the
range of the invariant (gen(M), dec(M)), which is mostly governed by the
inequality dec(M) leq C^{gen(M)}.