A History of Interactions between Logic and Number Theory

I concentrate on logic and number theory, and the evolution of this interaction. The period under review is 1929-2003, beginning with Presburger’s work [89]) on the metamathematics of addition and order on Z. Presburger’s work was published two years before the dramatic discoveries of Gödel, which revealed the ubiquity of undecidability and incompleteness in global arithmetic. Presburger showed that a certain core mathematical structure is non-Gödelian, and it is to be stressed that even now, seventy five years later, one is learning interesting new things about definitions in this structure [84]). Moreover, we know now that the p-adic fields are uniformly non-Gödelian, and we understand that this depends mainly on how their definable relations are built over those of Presburger. Since 1964 it has been evident that the local/global distinction in number theory corresponds to a non-Gödelian/Gödelian distinction in definability theory. Completing a structure tends to simplify definability theory, often changing a Gödelian situation to a non-Gödelian one. (There is one perplexing counterexample to this,the case of free groups and their profinite completions [14, 15]).