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On VC-minimal theories and variants

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Abstract

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN, 46556, USA

    Vincent Guingona

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Michael C. Laskowski

Authors
  1. Vincent Guingona
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  2. Michael C. Laskowski
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Corresponding author

Correspondence to Vincent Guingona.

Additional information

Vincent Guingona and Michael C. Laskowski were partially supported by Laskowski’s NSF Grants DMS-0600217 and 0901336.

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Guingona, V., Laskowski, M.C. On VC-minimal theories and variants. Arch. Math. Logic 52, 743–758 (2013). https://doi.org/10.1007/s00153-013-0341-z

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  • DOI: https://doi.org/10.1007/s00153-013-0341-z

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