 Continuum many one-types of the integer ring

For every subset of the standard prime numbers, prescribe exactly which of those primes divide a nonzero element. Every finite prescription is realized by the product of its finitely many positive divisibility requirements. Compactness theorem extends each prescription to a complete one-type of the complete theory of the integer ring. Different subsets give incompatible partial types. The countable first-order language supplies the matching upper bound.