Computable dimension for ordered fields

Abstract

The computable dimension of a structure counts the number of computable copies up to computable isomorphism. In this paper, we consider the possible computable dimensions for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable, and thus have computable dimension 1. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also such fields which are computably categorical. Finally, we show that 1 is the only possible finite computable dimension for any computable archimedean field.

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Correspondence to Oscar Levin.

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Levin, O. Computable dimension for ordered fields. Arch. Math. Logic 55, 519–534 (2016). https://doi.org/10.1007/s00153-016-0478-7

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Keywords

  • Computable dimension
  • Computable ordered fields
  • Computably categorical ordered fields
  • Effective algebra

Mathematics Subject Classification

  • 03D45
  • 03C57
  • 12J15
  • 12L12