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

Published online by Cambridge University Press:  12 March 2014

Ricardo Bianconi
Ricardo Bianconi*
Affiliation:
IME-USP, Caixa Postal 66281, Cep 05389-970, São Paulo, SP, Brazil, E-mail: bianconi@ime.usp.br
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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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1173 - 1178
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Ax, J., On Schanuel's conjectures, Annals of Mathematics, vol. 93 (1971), pp. 252–268.CrossRefGoogle Scholar
[2]Bianconi, R., On sets ∀-definable from Pfaffian functions, this Journal, vol. 57 (1992), pp. 688–697.Google Scholar
[3]Bianconi, R., Some model theory for the reals with analytic functions, Proceedings of the Tenth Brazilian Conference on Mathematical Logic, EBL93, Coleção CLE (Brazil), vol. 14, UNICAMP, 1995, pp. 61–71.Google Scholar
[4]Macintyre, A., Notes on exponentiation, 1984, mimeographed notes from a course given at Urbana, Ill.Google Scholar
[5]Macintyre, A., Schanuel's conjecture and free exponential rings, Annals of Pure and Applied Logic, vol. 51 (1991), pp. 241–246.CrossRefGoogle Scholar
[6]van den Dries, L., A generalization of Tarski-Seidenberg theorem and some nondefinability results, Bulletin of the AMS, vol. 15 (1986), pp. 189–193.CrossRefGoogle Scholar
[7]van den Dries, L., On the elementary theory of restricted elementary functions, this Journal, vol. 53 (1988), pp. 796–808.Google Scholar
[8]van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel Journal of Mathematics, vol. 85 (1994), pp. 19–58.CrossRefGoogle Scholar
[9]Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by Pfaffian functions and the exponential function, to appear.Google Scholar