Disjunctions with stopping condition

TL;DR

This paper develops a new analytical tool for models of the compositional truth theory over Peano Arithmetic, providing novel proofs of properties related to model saturation and truth predicates, including limitations on extending partial truth predicates.

Contribution

It introduces a new tool for analyzing models of $ extnormal{CT}^-$, offers a new proof of Lachlan's theorem, and constructs a partial truth predicate that cannot be extended to a full one.

Findings

Models of $ extnormal{PA}$ are recursively saturated.

All models of $ extnormal{CT}^-$ have a partial inductive truth predicate.

Existence of a partial truth predicate that cannot be extended to a full one.

Abstract

We introduce a tool for analysing models of $\textnormal{CT}^-$, the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan's theorem that arithmetical part of models of $\textnormal{PA}$ are recursively saturated. We use this tool to provide a new proof that all models of $\textnormal{CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for formulae from a nonstandard cut which cannot be extended to a full truth predicate satisfying $\textnormal{CT}^-$.