  We study random families of subsets of $\mathbb{N}$ that are similar to
exchangeable random partitions, but do not require constituent sets to be
disjoint: Each element of ${\mathbb{N}}$ may be contained in multiple subsets.
One class of such objects, known as Indian buffet processes, has become a
popular tool in machine learning. Based on an equivalence between Indian buffet
and scale-invariant Poisson processes, we identify a random scaling variable
whose role is similar to that played in exchangeable partition models by the
total mass of a random measure. Analogous to the construction of exchangeable
partitions from normalized subordinators, random families of sets can be
constructed from randomly scaled subordinators. Coupling to a heavy-tailed
scaling variable induces a power law on the number of sets containing the first
$n$ elements. Several examples, with properties desirable in applications, are
derived explicitly. A relationship to exchangeable partitions is made precise
as a correspondence between scaled subordinators and Poisson-Kingman measures,
generalizing a result of Arratia, Barbour and Tavare on scale-invariant
processes.

Probability