Definable sets in weak Presburger arithmetic

Presburger arithmetic is the fragment of arithmetic concerning integers with addition and order. Presburger’s supervisor considered the decidability of this fragment too modest a result to deserve a Ph.D. degree and he accepted it only as a Master’s Thesis in 1928. Looking at the number of citations, we may say that history revised this depreciative judgment long ago. There still remains, at least as far as we can see, some confusion concerning the definition itself of the structure: is the domain Z or N? Must we take the order relation or not? (The main popular mathematical websites disagree on this respect). The original paper deals with the additive group of positive and negative integers with no binary relation, but in a final remark of the communication, the author asserts that the same result, to wit quantifier elimination, holds on the structure of the “whole” integers, i.e., the natural numbers with the binary relation <. In 7, which is the main reference on the subject, Presburger arithmetic is defined as the elementary theory of integers with equality, addition, having 0 and 1 as constant symbols and < as binary pred-