On the Computability of Ordered Fields

Abstract

In this paper we develop general techniques for structures of computable real numbers generated by classes of total computable (recursive) functions with special restrictions on basic operations in order to investigate the following problems:
whether a generated structure is a real closed field and whether there exists a computable presentation of a generated structure.
We prove a series of theorems that lead to the result that there are no computable presentations
neither for polynomial time computable no even for $\mathcal{E}_n$-computable real numbers, where $\mathcal{E}_n$ is a level in Grzegorczyk hierarchy, $n\geq 2$.
We also propose a criterion of computable presentability of an archimedean ordered field.