Recursively saturated nonstandard models of arithmetic

Abstract

Through the ability of arithmetic to partially define truth and the ability of infinite integers to simulate limit processes, nonstandard models of arithmetic automatically have a certain amount of saturation: Any encodable partial type whose formulae all fall into the domain of applicability of a truth definition must, by finite satisfiability and Overspill, be nonstandard-finitely satisfiable—whence realized. This fact was first exploited by A. Robinson, who in Robinson [1963] cited the unrealizability in a given model of a certain encodable partial type to prove Tarski's Theorem on the Undefinability of Truth. A decade later, H. Friedman brought this phenomenon to the public's attention by using it to establish impressive embeddability criteria for countable nonstandard models of arithmetic. Subsequently, Wilkie considered models expandable to “strong theories” and, among such models, complemented Friedman's embeddability criteria with elementary embeddability and isomorphism criteria. Oddly enough, the fact that some kind of saturation property was being employed was not explicitly acknowledged in any of this work.It is in the unpublished dissertation of Wilmers that these submerged saturation properties first surfaced. [As I have only seen accounts of it (most notably Murawski [1976/1977]) and not the dissertation itself, what I have to say about it will not quite be accurate. (Indeed, the referee has refuted, without providing alternate information, every conjecture I have made about the contents of this thesis.)