Abstract

Let be the set of nonnegative integers. We show the two following facts about Presburger's arithmetic:1. 1. Let . If L is not definable in , + then there is an definable in , such that there is no bound on the distance between two consecutive elements of L′. and2. 2. is definable in , + if and only if every subset of which is definable in is definable in , +. These two Theorems are of independent interest but we will get from them new proofs of Cobham's and Semenov's Theorems ; Semenov's Theorem is:Let k and l be multiplicatively independent . If is definable in and in then L is recognizable . Here Vm is the function which sends a nonzero natural number to the greatest power of m dividing it