Arithmetic Saturation and Pathological Satisfaction

Abstract: A classic result in models of arithmetic states that countable models of PA are recursively saturated if and only if they possess a "full satisfaction class". A satisfaction class is a set of pairs (phi, alpha), where phi is a code for a formula in the sense of the model, and alpha is an assignment for that formula, which extends the "standard" satisfaction relation, and satisfies Tarksi's compositional rules for satisfaction. Recently, there has been work on so-called pathological satisfaction classes: satisfaction classes which exhibit certain pathologies, like, for example, making sentences of the form "(0 = 1) or (0 = 1) or ... or (0 =1)" of nonstandard length true. We study these pathologies, and find a surprising relationship between the question of determining which sets can be defined using certain pathologies, and a stronger notion of saturation, arithmetic saturation. This is joint work with Mateusz Łełyk, based heavily on unpublished work by Jim Schmerl.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic