New distance for any finite sets, half the Hamming distance

In this paper, we introduce a new, previously unknown, distance (i.e.,
a new metric) in a set whose elements are some other (any)
finite sets. It is proved that with such a metric the
set under consideration is a metric space. A direct relationship is
established between this distance and the Hamming distance: it is
exactly two times smaller than the Hamming distance and it is much
easier to calculate it. As an application, the set of natural numbers is
considered as a discrete metric space with a new metric introduced, and
a new metric criterion for the primality of a natural
number is established. This is the first metric criterion
in the history of mathematics for a natural number to be prime.