Satisfaction classes in nonstandard models of first-order arithmetic

TL;DR

This paper explores the properties and construction of satisfaction classes in nonstandard models of arithmetic, extending existing results and introducing new notions like free satisfaction classes and valuation concepts.

Contribution

It provides a detailed proof of the characterization of satisfaction classes in countable models of PA and introduces new concepts such as free satisfaction classes and valuation of nonstandard terms.

Findings

Countable models of PA admit satisfaction classes iff they are recursively saturated.

Introduction of free satisfaction classes free of existential assumptions.

Extensions of M-logic to remove pathologies are proposed, with some open questions on their consistency.

Abstract

A satisfaction class is a set of nonstandard sentences respecting Tarski's truth definition. We are mainly interested in full satisfaction classes, i.e., satisfaction classes which decides all nonstandard sentences. Kotlarski, Krajewski and Lachlan proved in 1981 that a countable model of PA admits a satisfaction class if and only if it is recursively saturated. A proof of this fact is presented in detail in such a way that it is adaptable to a language with function symbols. The idea that a satisfaction class can only see finitely deep in a formula is extended to terms. The definition gives rise to new notions of valuations of nonstandard terms; these are investigated. The notion of a free satisfaction class is introduced, it is a satisfaction class free of existential assumptions on nonstandard terms. It is well known that pathologies arise in some satisfaction classes. Ideas of how to remove those are presented in the last chapter. This is done mainly by adding inference rules to M-logic. The consistency of many of these extensions is left as an open question.