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 [14] . Another stretching theorem of Finkel and Ressayre implies that one can decide, for a given local sentence ϕ and an ordinal α < ω ω , whether ϕ has a model of order type α. This result is very similar to Büchi's one who proved that the monadic second order

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... ory of the structure (α, <), for a countable ordinal α, is decidable. It is in fact an extension of that result, as shown in [6] by considering the expressive power of monadic sentences and of local sentences over languages of words of length α. The aim of this paper is twofold. We wish first to attract the reader's attention on these powerful decidability results proved using methods of model theory and which should find some applications in computer science and we prove also here two 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 ωα (the α-th cardinal), whether a local sentence ϕ has a model of order type ωα. 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 ω2 is not determined by the axiomatic system ZF C + GCH, where GCH is the generalized continuum hypothesis.