Abstract

A structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; in particular we show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author: (i) decidability of the spectrum $\operatorname{Sp}(\varphi)$ of a local sentence φ, below ω ω ; where $\operatorname{Sp}(\varphi)$ is the set of ordinals α such that φ has a model of order type α (ii) proof that $\operatorname{beth}_\omega = \sup\{\operatorname{Sp}(\varphi): \varphi \text{local sentence with a bounded spectrum}\}$ (iii) existence of a local sentence φ such that $\operatorname{Sp}(\varphi)$ contains all infinite ordinals except the inaccessible cardinals