On Decidability Properties of Local Sentences

TL;DR

This paper explores the decidability of local first-order sentences, proving new results for unary function symbols and demonstrating limitations in the general case under certain set-theoretic assumptions.

Contribution

It introduces a new decidability result for local sentences with unary functions and shows that this result cannot be extended to all local sentences, highlighting set-theoretic limitations.

Findings

Decidability of local sentences with unary functions for regular cardinals

Limitations of extending decidability results to general local sentences

Dependence on set-theoretic assumptions like inaccessible cardinals

Abstract

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences. We prove here several additional results on local sentences. The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal k whether a local sentence phi has a model of order type k. Secondly we show that this result can not be extended to the general case. Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type omega_2 is not determined by the axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesis