Geodesic
Some presentations of the real number field
Algebra i logika, Tome 51 (2012) no. 1, pp. 96-128.

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It is proved that every two $\Sigma$-presentations of an ordered field $\mathbb R$ of reals over $\mathbb{HF(R)}$, whose universes are subsets of $\mathbb R$, are mutually $\Sigma$-isomorphic. As a consequence, for a series of functions $f\colon\mathbb R\to\mathbb R$ (e.g., $\exp$, $\sin$, $\cos$, $\ln$), it is stated that the structure $\mathbb R=\langle R,+,\times,,0,1,f\rangle$ lacks such $\Sigma$-presentations over $\mathbb{HF(R)}$.
Keywords: $\Sigma$-presentation, ordered field of reals. 
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A. S. Morozov. Some presentations of the real number field. Algebra i logika, Tome 51 (2012) no. 1, pp. 96-128. http://geodesic.mathdoc.fr/item/AL_2012_51_1_a4/

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