Infinite Partial Sumsets in the Primes

It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture).  Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 less than a_2 less than ... and b_1 less than b_2 less than ... such that a_i + b_j is prime for all i less than j.  Equivalently, the primes are not "translation-finite" in the sense of Ruppert.  As an application of these methods we show that the orbit closure of the primes is uncountable.