  In this paper we develop general techniques for classes of computable real
numbers generated by subsets of total computable (recursive functions) with
special restrictions on basic operations in order to investigate the following
problems: whether a generated class is a real closed field and whether there
exists a computable presentation of a generated class. 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 $E_n$-computable real
numbers, where $E_n$ is a level in Grzegorczyk hierarchy, $n \geq 2$. We also
propose a criterion of computable presentability of an archimedean ordered
field.
