In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation (...) of divisibility | (wherex|ymeansxdividesy). (shrink)

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We (...) also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)