On $d$-finiteness in continuous structures

Itaï Ben Yaacov ¹   ; Alexander Usvyatsov ²

¹ Université de Lyon Université Lyon 1 Institut Camille Jordan, CNRS, UMR 5208 43 boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France
² Mathematics Department UCLA Box 951555OC Los Angeles, CA 90095-1555, U.S.A.

Fundamenta Mathematicae, Tome 194 (2007) no. 1, pp. 67-88

Résumé

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. The notion of a $d$-finite tuple attempts to capture some aspects of the classical finite tuple behaviour. We show that many classical results involving finite tuples are valid in continuous logic upon replacing “finite” with “$d$-finite”. Other results, such as Vaught's no two models theorem and Lachlan's theorem on the number of countable models of a superstable theory are proved under the assumption of enough (uniformly) $d$-finite tuples.

Détail

DOI : 10.4064/fm194-1-4

Keywords: observe certain classical results first order model theory fail context continuous first order logic argue happens since finite tuples continuous structure may behave infinite tuples classical model theory notion d finite tuple attempts capture aspects classical finite tuple behaviour many classical results involving finite tuples valid continuous logic replacing finite d finite other results vaughts models theorem lachlans theorem number countable models superstable theory proved under assumption enough uniformly d finite tuples