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Miscellaneous

  • Frank O. Wagner
Part of the Mathematics and Its Applications book series (MAIA, volume 503)

Abstract

Note that smallness is preserved under naming finitely many parameters, as n-types over a parameter set of size m induce (m + n)-types over Ø. Small theories are of particular interest in connection with Vaught’s conjecture, which states that every countable theory should either have countably or continuum many countable models, up to isomorphism; this has so far been shown only for superstable theories of finite SU-rank [18]. Since a countable model can only realize countably many types and S (Ø) is either countable or has size continuum, it follows that a theory with fewer than continuum many non-isomorphic countable models must be small. Further examples of small theories are ω-categorical theories, where the Ryll-Nardzewski Theorem states that S n (Ø) is finite for all n < ω. These shall be studied later in section 6.2.

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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Frank O. Wagner
    • 1
  1. 1.Institut Girard DesarguesUniversité Claude Bernard (Lyon-1)VilleurbanneFrance

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