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Stretchings

Published online by Cambridge University Press:  12 March 2014

O. Finkel  and
J. P. Ressayre
J. P. Ressayre
Affiliation:
Equipe de Logique Mathématique (URA 753, CNRS), Université Paris7, 2, Place Jussieu - Case 7012, 75251 Paris Cedex 05, France, E-mail: Ressayre@logique.Jussieu.fr
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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 Sp(φ) of a local sentence φ, below ωω; where Sp(φ) is the set of ordinals α such that φ has a model of order type α

(ii) proof that bethω = sup{Sp(φ): φ local sentence with a bounded spectrum}

(iii) existence of a local sentence φ such that Sp(φ) contains all infinite ordinals except the inaccessible cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 563 - 585
Copyright
Copyright © Association for Symbolic Logic 1996

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References

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