Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T08:52:32.997Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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