On the elementary theory of restricted elementary functions

Abstract

As a contribution to definability theory in the spirit of Tarski's classical work on (R, <, 0, 1, +, ·) we extend here part of his results to the structureHere exp ∣[0, 1] and sin ∣[0, π] are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on R. The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in RRE.Let (RRE, constants) be the expansion of RRE obtained by adding a name for each real number to the language. We can now formulate our main result as follows.Theorem. (RRE, constants) is strongly model-complete.This means that every formula ϕ(X1, …, Xm) in the natural language of (RRE, constants) is equivalent to an existential formulawith the extra property that for each x ∈ Rm such that ϕ(x) is true in RRE there is exactly one y ∈ Rn such that ψ(x, y) is true in RRE. (Here ψ is quantifier free.)