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)

This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is first-order definable over the class of finite directed graphs and that this class admits a first-order definable global linear order. We apply this last result to show that FO = FO.