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Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function

  • Ricardo Bianconi (a1)
  • DOI: http://dx.doi.org/10.2307/2275634
  • Published online: 01 March 2014
Abstract
Abstract

We prove that no restriction of the sine function to any (open and nonempty) interval is definable in 〈R, +, ·, ×, <, exp, constants〉, and that no restriction of the exponential function to an (open and nonempty) interval is definable in 〈R, +, ·, <, sin0, constants〉, where sin0(x) = sin(x) for x ∈ [—π, π], and sin0(x) = 0 for all x ∉ [—π, π].

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[1]J. Ax , On Schanuel's conjectures, Annals of Mathematics, vol. 93 (1971), pp. 252268.

[5]A. Macintyre , Schanuel's conjecture and free exponential rings, Annals of Pure and Applied Logic, vol. 51 (1991), pp. 241246.

[6]L. van den Dries , A generalization of Tarski-Seidenberg theorem and some nondefinability results, Bulletin of the AMS, vol. 15 (1986), pp. 189193.

[8]L. van den Dries and C. Miller , On the real exponential field with restricted analytic functions, Israel Journal of Mathematics, vol. 85 (1994), pp. 1958.

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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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