In the present paper we investigate algorithmic properties of fields. We prove
that axioms of formally real fields for the field
R
of reals and axioms of fields of
characteristic zero for the field
C
of complex
numbers, give the complete characterization of algorithmic properties. By
Kfoury’s theorem programs which define total functions over
R
or
C
are
effectively equivalent to loop-free programs. Examples of programmable and
nonprogrammable functions and relations over
R
and
C
are given. In the case of ordered reals
the axioms of Archimedean ordered fields completely characterize algorithmic
properties. We show how to use the equivalent version of Archimed’s axiom (the
exhaustion rule) in order to prove formally the correctness of some iterative
numerical algorithms.
