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On the elementary theory of restricted elementary functions. (English) Zbl 0698.03023
As a contribution to definability theory in the spirit of Tarski’s classical work on \(({\mathbb{R}},<,0,1,+,\cdot)\) we extend here part of his results to the structure \[ {\mathbb{R}}^{RE}=({\mathbb{R}},<,0,1,+,\cdot,\exp |_{[0,1]},\sin |_{[0,\pi]}). \] Here \(\exp |_{[0,1]}\) and \(\sin |_{[0,\pi]}\) 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 \({\mathbb{R}}\). The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in \({\mathbb{R}}^{RE}.\)
Let \(({\mathbb{R}}^{RE}\), constants) be the expansion of \({\mathbb{R}}^{RE}\) obtained by adding a name for each real number to the language. We can now formulate our main result as follows. Theorem. \(({\mathbb{R}}^{RE}\), constants) is strongly model-complete.
This means that every formula \(\phi (X_ 1,...,X_ m)\) in the natural language of \(({\mathbb{R}}^{RE}\), constants) is equivalent to an existential formula \(\exists Y_ 1...Y_ n\) \(\psi (X_ 1,...,X_ m,Y_ 1,...,Y_ n)\) with the extra property that for each \(x\in {\mathbb{R}}^ m\) such that \(\phi\) (x) is true in \({\mathbb{R}}^{RE}\) there is exactly one \(y\in {\mathbb{R}}^ n\) such that \(\psi\) (x,y) is true in \({\mathbb{R}}^{RE}\). (Here \(\psi\) is quantifier free.) In other words, strong model- completeness is the next best thing after quantifier elimination, and is of course stronger than model-completeness.

MSC:
03C40 Interpolation, preservation, definability
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References:
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