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.