Definability and decision problems in arithmetic

It is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation of divisibility | (where x|y means x divides y).

Abstract

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 successor S (where Sa = a + 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation of divisibility | (where x|y means x divides y ).