On $d$-finiteness in continuous structures

Volume 194 / 2007

Itaï Ben Yaacov, Alexander Usvyatsov Fundamenta Mathematicae 194 (2007), 67-88 MSC: 03C90, 03C45, 03C95, 03C35. DOI: 10.4064/fm194-1-4


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.


  • Itaï Ben YaacovUniversité de Lyon
    Université Lyon 1
    Institut Camille Jordan, CNRS, UMR 5208
    43 boulevard du 11 novembre 1918
    69622 Villeurbanne Cedex, France
  • Alexander UsvyatsovMathematics Department
    Box 951555OC
    Los Angeles, CA 90095-1555, U.S.A.

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