Nonstandard arithmetic and recursive comprehension

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006) 100–125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory RCA 0 , has a natural nonstandard counterpart. But the counterpart ∗ RCA 0 of RCA 0 has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this paper we find another nonstandard counterpart, ∗ RCA 0 ′ , that does imply the Standard Part Principle.