Ordinal computability uses ordinals instead of natural numbers in abstract machines like register or  Turing  machines. We give an overview of the computational strengths of  ***  - β  -machines, where  ***  and  β  bound the time axis and the space axis of some machine model. The spectrum ranges from classical  Turing  computability to ***-***-computability which corresponds to  Godel  's model of constructible sets. To illustrate some typical techniques we prove a new result on Infinite Time Register Machines (= ***- ***  -register machines) which were introduced in [6]: a real number  x  ***  ***  2 is computable by an Infinite Time Register Machine iff it is  Turing  computable from some finitely iterated hyperjump 0( n  ).
['Peter Koepke']
Ordinal Computability