  In this paper we study the system of sets of lengths of non-finitely
generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of
$\mathbb{Q}_{\ge 0}$). We begin by presenting a BF-monoid $M$ with full system
of sets of lengths, which means that for each subset $S$ of $\mathbb{Z}_{\ge
2}$ there exists an element $x \in M$ whose set of lengths $\mathsf{L}(x)$ is
$S$. It is well known that systems of sets of lengths do not characterize
numerical monoids. Here, we prove that systems of sets of lengths do not
characterize non-finitely generated atomic Puiseux monoids. In a recent paper,
Geroldinger and Schmid found the intersection of systems of sets of lengths of
numerical monoids. Motivated by this, we extend their result to the setting of
atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux
monoid $P = \langle 1/p \mid p \ \text{is prime} \rangle$ with the Goldbach's
conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of
Goldbach's numbers.
