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On the elementary theory of restricted elementary functions

Published online by Cambridge University Press:  12 March 2014

Lou van den Dries
Lou van den Dries*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
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Extract

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 structure

Here 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 formula

with 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.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 796 - 808
Copyright
Copyright © Association for Symbolic Logic 1988

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References

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