  We extend in a natural way the operation of Turing machines to infinite
ordinal time, and investigate the resulting supertask theory of computability
and decidability on the reals. The resulting computability theory leads to a
notion of computation on the reals and concepts of decidability and
semi-decidability for sets of reals as well as individual reals. Every Pi^1_1
set, for example, is decidable by such machines, and the semi-decidable sets
form a portion of the Delta^1_2 sets. Our oracle concept leads to a notion of
relative computability for reals and sets of reals and a rich degree structure,
stratified by two natural jump operators.
