  A new number system, the set of the non-Dedekindian numbers, is introduced
and characterized axiomatically. It is then proved that any hypercontinous
hyperreal number system is strictly included in the set of the Non-Dedekindian
Numbers. The notion of algorithmic-randomness is then extended to
non-Dedekindian numbers. As a particular case, the notion of algorithmic
randomness for the particular hyperreal number system of Non-Standard Analysis
is explicitly analyzed.
