Models of Weak Theories of Truth

TL;DR

This paper compares the strength of various weak truth theories over PA using model-theoretic methods, establishing inclusion relations among classes of models and showing that models of CT− can be expanded to UTB, clarifying interpretability relations.

Contribution

It introduces a model-theoretic framework for comparing weak truth theories and proves that models of CT− can be expanded to models of UTB, answering a key interpretability question.

Findings

Models of PA admit expansions to various truth theories in a hierarchy.

Every model of PA with a CT− expansion can also be expanded to UTB.

UTB is not relatively interpretable in TB, resolving an open question.

Abstract

In the following paper we propose a model-theoretical way of comparing the "strength" of various truth theories which are conservative over PA. Let $\mathfrak{Th}$ denote the class of models of PA which admit an expansion to a model of theory Th. We show (combining some well known results and original ideas) that $$\mathfrak{PA}\supset \mathfrak{TB}\supset \mathfrak{RS}\supset \mathfrak{UTB}\supseteq\mathfrak{CT^-},$$ where $\mathfrak{PA}$ denotes simply the class of all models of PA and $\mathfrak{RS}$ denotes the class of recursively saturated models of PA. Our main original result is that every model of PA which admits an expansion to a model of CT${}^-$, admits also an expanion to a model of UTB. Moreover, as a corollary to one of the results we conclude that UTB is not relatively interpretable in TB, thus answering the question of Fujimoto.