On some representations of the real number field

Abstract

Work on problems that we are coming up with began with an inquiry into whether the structure 〈R,+,×, <, 0, 1, exp〉, where R = 〈R,+,×, <, 0, 1〉 is an ordered field of real numbers, is Σ-definable over HF(R). In terms of theoretical programming, the question above can be informally restated as follows. Assume that there exists some hypothetical programming language which admits of a faithful realization of real numbers, with +, ×, and a strict order relation <. Suppose that this language has rich possibilities for constructing abstract data types. In other words, the language contains a collection of tools which allows us to determine new abstract structures based on available ones and then program calculations over just these new structures. Usually, in realizing a function exp(x) = ex, we have to use its expansion into a series and content ourselves with more or less exact approximations. Is it possible that this system will allow for defining in it a new structure, classically isomorphic to an ordered field of reals, in which the function exp will be computed accurately? Also of interest are similar questions for ln, sin, cos, and other functions. The intuitively correct answer being ‘no’ is in fact not so obvious: one of our results says that the restriction of the structure in question to 〈R;<, exp〉 already has a rather simple Σrepresentation over HF(R). In addition, it is shown that the algebraic structure 〈R,+,×, <, 0, 1, f〉, where f is one of the functions exp, sin, cos, or ln, has no Σ-representation over HF(R), whose universe is a subset of R. The statements proved can be used to uniformly derive analogous results for some other functions as well. Such a possibility is presented by the following: