  Natural collectives, despite comprising individuals who may not know their
numerosity, can exhibit behaviors that depend sensitively on it. This paper
proves that the collective behavior of number-oblivious individuals can even
have a critical numerosity, above and below which it qualitatively differs. We
formalize the concept of critical numerosity in terms of a family of zero--one
laws and introduce a model of collective motion, called chain activation and
transport (CAT), that has one.
  CAT describes the collective motion of $n \geq 2$ individuals as a Markov
chain that rearranges $n$-element subsets of the $d$-dimensional grid, $m < n$
elements at a time. According to the individuals' dynamics, with each step, CAT
removes $m$ elements from the set and then progressively adds $m$ elements to
the boundary of what remains, in a way that favors the consecutive addition and
removal of nearby elements. This paper proves that, if $d \geq 3$, then CAT has
a critical numerosity of $n_c = 2m+2$ with respect to the behavior of its
diameter. Specifically, if $n < n_c$, then the elements form one "cluster," the
diameter of which has an a.s.--finite limit infimum. However, if $n \geq n_c$,
then there is an a.s.--finite time at which the set consists of clusters of
between $m+1$ and $2m+1$ elements, and forever after which these clusters grow
apart, resulting in unchecked diameter growth.
  The existence of critical numerosities means that collectives can exhibit
"phase transitions" that are governed purely by their numerosity and not, for
example, their density or the strength of their interactions. This fact
challenges prevalent beliefs about collective behavior and suggests new
functionality for programmable matter. More broadly, it demonstrates an
opportunity to explore the possible behaviors collectives through the study of
random processes that rearrange sets.
