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

Abstract

AbstractWe 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].