Local collection scheme and end-extensions of models of compositional truth

TL;DR

This paper introduces a local collection principle for compositional truth predicates, showing it is conservative over classical theories and analyzing model extensions and collection schemes in the context of truth theories.

Contribution

It presents a new local collection principle for compositional truth, demonstrating its conservativity and analyzing end-extensions of models in this framework.

Findings

Local collection is conservative over classical compositional truth theories.

Arguments using collection do not suffice to prove the truth of the conclusion.

Analysis of end-extensions of models of compositional truth and collection schemes.

Abstract

We introduce a principle of local collection for compositional truth predicates and show that it is conservative over the classically compositional theory of truth in the arithmetical setting. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the case of induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate.