Some presentations of the real number field

Abstract

It is proved that every two Σ-presentations of an ordered field $ \mathbb{R} $ of reals over $ \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) $ , whose universes are subsets of $ \mathbb{R} $ , are mutually Σ-isomorphic. As a consequence, for a series of functions $ f:\mathbb{R} \to \mathbb{R} $ (e.g., exp, sin, cos, ln), it is stated that the structure $ \mathbb{R} $ = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over $ \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) $ .