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Expansions of the real field by open sets: definability versus interpretability

  • Harvey Friedman (a1), Krzysztof Kurdyka (a2), Chris Miller (a3) and Patrick Speissegger (a4)
  • DOI: http://dx.doi.org/10.2178/jsl/1286198148
  • Published online: 01 March 2014
Abstract
Abstract

An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]Harvey Friedman and Chris Miller , Expansions of o-minimal structures by sparse sets, Fundamenta Mathematicae, vol. 167 (2001), no. 1, pp. 5564.

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[11]Chris Miller , Avoiding the projective hierarchy in expansions of the real field by sequences, Proceedings of the American Mathematical Society, vol. 134 (2006), no. 5, pp. 14831493 (electronic).

[13]Chris Miller and James Tyne , Expansions of o-minimal structures by iteration sequences, Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 1, pp. 9399 (electronic).

[16]Lou van den Dries , Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.

[17]Lou van den Dries and Chris Miller , Geometric categories and o-minimal structures, Duke Mathematical Journal, vol. 84 (1996), no. 2, pp. 497540.

[18]Yosef Yomdin and Georges Comte , Tame geometry with application in smooth analysis, Lecture Notes in Mathematics, vol. 1834, Springer-Verlag, Berlin, 2004.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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